A000009 -id:A000009

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A047966

a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

+20
143

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004

Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018

Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). - Gary W. Adamson, Jun 08 2009

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

FORMULA

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003

G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018

a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

EXAMPLE

The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - Gus Wiseman, Apr 16 2018

MAPLE

with(numtheory):

b:= proc(n) option remember; `if`(n=0, 1, add(add(

`if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)

end:

a:= n-> add(b(d), d=divisors(n)):

seq(a(n), n=1..100); # Alois P. Heinz, Jul 11 2016

MATHEMATICA

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)

Table[DivisorSum[n, PartitionsQ], {n, 20}] (* Gus Wiseman, Apr 16 2018 *)

PROG

(PARI)

N = 66; q='q+O('q^N);

D(q)=eta(q^2)/eta(q); \\ A000009

Vec( sum(e=1, N, D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

CROSSREFS

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A089259

Expansion of Product_{m>=1} 1/(1-x^m)^A000009(m).

+20
74

1, 1, 2, 4, 7, 12, 22, 36, 61, 101, 166, 267, 433, 686, 1088, 1709, 2671, 4140, 6403, 9824, 15028, 22864, 34657, 52288, 78646, 117784, 175865, 261657, 388145, 573936, 846377, 1244475, 1825170, 2669776, 3895833, 5671127, 8236945, 11936594, 17261557, 24909756

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Number of complete set partitions of the integer partitions of n. This is the Euler transform of A000009. If we change the combstruct command from unlabeled to labeled, then we get A000258. - Thomas Wieder, Aug 01 2008

Number of set multipartitions (multisets of sets) of integer partitions of n. Also a(n) < A270995(n) for n>5. - Gus Wiseman, Apr 10 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

EXAMPLE

From Gus Wiseman, Oct 22 2018: (Start)

The a(6) = 22 set multipartitions of integer partitions of 6:

(6) (15) (123) (12)(12) (1)(1)(1)(12) (1)(1)(1)(1)(1)(1)

(24) (1)(14) (1)(1)(13) (1)(1)(1)(1)(2)

(1)(5) (1)(23) (1)(2)(12)

(2)(4) (2)(13) (1)(1)(1)(3)

(3)(3) (3)(12) (1)(1)(2)(2)

(1)(1)(4)

(1)(2)(3)

(2)(2)(2)

(End)

MAPLE

with(combstruct): A089259:= [H, {H=Set(T, card>=1), T=PowerSet (Sequence (Z, card>=1), card>=1)}, unlabeled]; 1, seq (count (A089259, size=j), j=1..16); # Thomas Wieder, Aug 01 2008

# second Maple program:

with(numtheory):

b:= proc(n, i)

if n<0 or n>i*(i+1)/2 then 0

elif n=0 then 1

elif i<1 then 0

else b(n, i):= b(n-i, i-1) +b(n, i-1)

end:

a:= proc(n) option remember; `if` (n=0, 1,

add(add(d* b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..100); # Alois P. Heinz, Nov 11, 2011

MATHEMATICA

max = 40; CoefficientList[Series[Product[1/(1-x^m)^PartitionsQ[m], {m, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 24 2014 *)

b[n_, i_] := b[n, i] = Which[n<0 || n>i*(i+1)/2, 0, n == 0, 1, i<1, 0, True, b[n-i, i-1] + b[n, i-1]]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d* b[d, d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

PROG

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

seq(n)={concat([1], EulerT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)))} \\ Andrew Howroyd, Oct 26 2018

CROSSREFS

Row sums of A285229 and of A360763.

Cf. A000009, A001970, A049311, A050342, A056156, A068006, A089254, A116540, A218153, A270995, A296119, A318360.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 23 2003

STATUS

approved

A270995

Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

+20
71

1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The number of ways a number can be partitioned into not necessarily distinct parts and then each part is partitioned into distinct parts. Also a(n) > A089259(n) for n>5. - Gus Wiseman, Apr 10 2016

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers with weakly decreasing multiplicities. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 7 multiset partitions are:

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{2},{1,1}} {{1,1},{2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The weakly normal non-strict version is A055887.

The non-strict version is A063834.

The weakly normal version is A304969.

(End)

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

FORMULA

From Vaclav Kotesovec, Mar 28 2016: (Start)

a(n) ~ c * n^2 * 2^(n/3), where

c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0

c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1

c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2

(End)

EXAMPLE

a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.

The unordered version is A089259, non-strict A001970 (row-sums of A061260).

For compositions instead of partitions we have A304969, non-strict A055887.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by sum and length.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Mar 28 2016

STATUS

approved

A036469

Partial sums of A000009 (partitions into distinct parts).

+20
50

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88, 110, 137, 169, 207, 253, 307, 371, 447, 536, 640, 762, 904, 1069, 1261, 1483, 1739, 2035, 2375, 2765, 3213, 3725, 4310, 4978, 5738, 6602, 7584, 8697, 9957, 11383, 12993, 14809, 16857, 19161, 21751, 24661

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Also number of 1's in all partitions of n+1 into odd parts. Example: a(4)=7 because the partitions of 5 into odd parts are [5], [3,1,1], [1,1,1,1,1], having a total number of 7 1's. - Emeric Deutsch, Mar 29 2006

Convolved with A035363 = A000070. - Gary W. Adamson, Jun 09 2009

Equals row sums of triangle A166240. - Gary W. Adamson, Oct 09 2009

a(n) = if n <= 1 then A201377(1,n) else A201377(n,1). - Reinhard Zumkeller, Dec 02 2011

a(n) equals the sum of the parts of the form 2^k (k >= 0) in all partitions of n + 1 into distinct parts. Example: a(6) = 14. The partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] having sum over parts of the form 2^k equal to 1 + 2 + 4 + 4 + 2 + 1 = 14. - Peter Bala, Dec 01 2013

Number of partitions of the (n+1)-multiset {0,...,0,1} with n 0's into distinct multisets; a(3) = 5: 0|00|1, 00|01, 000|1, 0|001, 0001. Also number of factorizations of 3*2^n into distinct factors; a(3) = 5: 2*3*4, 4*6, 3*8, 2*12, 24. - Alois P. Heinz, Jul 30 2021

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.

A. V. Chekhonadskikh, Some classical number sequences in control system design, Siberian Electronic Mathematical Reports, Volume 14, p. 620-628. See Theorem 2.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 774

FORMULA

G.f.: 1/[(1-x)*product(1-x^(2j-1), j=1..infinity)]. - Emeric Deutsch, Mar 29 2006

a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)) * (1 + (18+13*Pi^2) / (48*Pi*sqrt(3*n)) + (2916 - 1404*Pi^2 + 121*Pi^4)/(13824*Pi^2*n)). - Vaclav Kotesovec, Feb 26 2015, updated Oct 26 2016

For n > 0, a(n) = A026906(n) + 1. - Vaclav Kotesovec, Oct 26 2016

Faster converging g.f.: A(x) = (1/(1 - x))*Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). - Peter Bala, Feb 02 2021

MAPLE

g:=1/(1-x)/product(1-x^(2*j-1), j=1..30): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..46); # Emeric Deutsch, Mar 29 2006

# second Maple program:

b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,

b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))

end:

a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:

seq(a(n), n=0..60); # Alois P. Heinz, Nov 21 2012

MATHEMATICA

CoefficientList[ Series[Product[(1 + t^i), {i, 1, Infinity}]/(1 - t), {t, 0, 46}], t] (* Geoffrey Critzer, May 16 2010 *)

b[n_, i_] := If[n == 0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, Min[n-i, i-1]]]]]; a[n_] := a[n] = b[n, n]+If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Accumulate[Table[PartitionsQ[n], {n, 0, 50}]] (* Vaclav Kotesovec, Oct 26 2016 *)

CROSSREFS

Cf. A000009, A265093.

Cf. A035363, A000070. - Gary W. Adamson, Jun 09 2009

Cf. A166240. - Gary W. Adamson, Oct 09 2009

Column k=1 of A346520.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A304969

Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

+20
46

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Invert transform of A000009.

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{1},{2,2}} {{1,1},{2,2}}

{{2},{1,1}} {{1},{2,2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{3},{2,2}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The non-strict version is A055887.

The strongly normal non-strict version is A063834.

The strongly normal version is A270995.

(End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2816

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Partition Function Q

Index entries for sequences related to partitions

Index entries for sequences related to compositions

FORMULA

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).

G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).

G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).

G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).

a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

EXAMPLE

From Gus Wiseman, Jul 31 2022: (Start)

a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:

((1)) ((2)) ((3)) ((4))

((1)(1)) ((21)) ((31))

((1)(2)) ((1)(3))

((2)(1)) ((2)(2))

((1)(1)(1)) ((3)(1))

((1)(21))

((21)(1))

((1)(1)(2))

((1)(2)(1))

((2)(1)(1))

((1)(1)(1)(1))

(End)

MAPLE

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

end:

a:= proc(n) option remember; `if`(n=0, 1,

add(b(j)*a(n-j), j=1..n))

end:

seq(a(n), n=0..40); # Alois P. Heinz, May 22 2018

MATHEMATICA

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]

a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

CROSSREFS

Cf. A050342, A055887, A067687, A081362, A279785, A299106.

Row sums of A308680.

The unordered version is A089259, non-strict A001970 (row-sums of A061260).

For partitions instead of compositions we have A270995, non-strict A063834.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by sum and length.

Cf. A279784.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 22 2018

STATUS

approved

A050342

Expansion of Product_{m>=1} (1+x^m)^A000009(m).

+20
35

1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

FORMULA

Weigh transform of A000009.

EXAMPLE

4=(4)=(3)+(1)=(3+1)=(2+1)+(1).

From Gus Wiseman, Oct 11 2018: (Start)

a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:

{{1},{2}} {{1},{3}} {{2,3}} {{2,4}}

{{1},{1,2}} {{1},{4}} {{1,2,3}}

{{2},{3}} {{1},{5}}

{{1},{1,3}} {{2},{4}}

{{2},{1,2}} {{1},{1,4}}

{{1},{2,3}}

{{2},{1,3}}

{{3},{1,2}}

{{1},{2},{3}}

{{1},{2},{1,2}}

(End)

MAPLE

g:= proc(n, i) option remember; `if`(n=0, 1,

`if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))

end:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))

end:

a:= n-> b(n, n):

seq(a(n), n=0..50); # Alois P. Heinz, May 19 2013

MATHEMATICA

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)

nn=10; Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k], {k, nn}], {x, 0, n}], {n, 0, nn}] (* Gus Wiseman, Oct 11 2018 *)

CROSSREFS

Cf. A050343-A050350, A089254.

Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.

Row sums of A330462 and of A360764.

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

A246867

Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

+20
31

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

LINKS

Alois P. Heinz, Rows n = 0..42, flattened

EXAMPLE

The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].

For n=0 the empty partition [] gives the empty product 1.

Triangle T(n,k) begins:

5, 6;

7, 10;

11, 14, 15;

13, 21, 22, 30;

17, 26, 33, 35, 42;

19, 34, 39, 55, 66, 70;

23, 38, 51, 65, 77, 78, 105, 110;

29, 46, 57, 85, 91, 102, 130, 154, 165, 210;

...

Corresponding triangle of strict integer partitions begins:

(1)

(2)

(3) (21)

(4) (31)

(5) (41) (32)

(6) (42) (51) (321)

(7) (61) (52) (43) (421)

(8) (71) (62) (53) (521) (431)

(9) (81) (72) (63) (54) (621) (432) (531). - Gus Wiseman, Feb 23 2018

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(

map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))

end:

T:= n-> sort(b(n$2))[]:

seq(T(n), n=0..14);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

CROSSREFS

Column k=1 gives: A008578(n+1).

Last elements of rows give: A246868.

Row sums give A147655.

Row lengths are: A000009.

Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

KEYWORD

nonn,tabf,look

AUTHOR

Alois P. Heinz, Sep 05 2014

STATUS

approved

A293467

a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

+20
24

1, 0, 0, -1, -3, -7, -14, -25, -41, -64, -100, -165, -294, -550, -1023, -1795, -2823, -3658, -2882, 2873, 20435, 62185, 148863, 314008, 613957, 1155794, 2175823, 4244026, 8753538, 19006490, 42471787, 95234575, 210395407, 453413866, 949508390, 1931939460

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Graph: a(n)/2^n (40000 terms)

MATHEMATICA

Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

CROSSREFS

Cf. A025147, A266232, A294467, A294468.

Cf. A218481, A294466, A095051.

The non-strict version is the absolute value of A281425; see A175804, A320590.

Up to sign, same as A380412. See A320591, A377285, A378970, A378971.

A000009 counts strict integer partitions, differences A087897.

Cf. A000041, A002865, A007442, A008284, A116608.

KEYWORD

sign

AUTHOR

Vaclav Kotesovec, Oct 09 2017

STATUS

approved

A035359

Number of partitions-into-distinct-parts of n (A000009) is a prime.

+20
23

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

LINKS

Table of n, a(n) for n=1..17.

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Eric Weisstein's World of Mathematics, Partition Function Q

Eric Weisstein's World of Mathematics, Partition Function Q-Congruences

EXAMPLE

From Gus Wiseman, Jan 13 2020: (Start)

Strict partitions of a(1) = 3 through a(4) = 7:

(3) (4) (5) (7)

(2,1) (3,1) (3,2) (4,3)

(4,1) (5,2)

(6,1)

(4,2,1)

(End)

MATHEMATICA

n = 1; A035359 = {}; While[n < 10^7, n++; If[ PrimeQ[ PartitionsQ[n]], Print[n]; AppendTo[A035359, n]]]; A035359 (* Jean-François Alcover, Oct 12 2011 *)

CROSSREFS

The non-strict version is A046063.

The version for powers of 2 instead of primes is A331022.

The version for factorizations instead of strict partitions is A330991.

The version for strict factorizations instead of strict partitions is A331201.

Cf. A000009, A051005, A056848, A265835.

KEYWORD

nonn,nice,hard,more

AUTHOR

Olivier Gérard

EXTENSIONS

More terms from Eric W. Weisstein

a(12) from Max Alekseyev, Jul 04 2009

a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015

a(15)-a(17) from Max Alekseyev, Jul 10 2015

STATUS

approved

A266232

Binomial transform of the number of partitions into distinct parts (A000009).

+20
23

1, 2, 4, 9, 21, 49, 114, 265, 615, 1422, 3272, 7493, 17090, 38850, 88065, 199097, 448953, 1009788, 2265642, 5071611, 11328395, 25254093, 56195143, 124829822, 276839061, 612991848, 1355268779, 2992016128, 6596222234, 14522634554, 31933047707, 70130243427

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then

Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where

g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).

Special cases:

p < 1/2, g(n) = 0

p = 1/2, g(n) = r^2/16

p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81

p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536

p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))

p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..3200

FORMULA

a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)).

G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - Ilya Gutkovskiy, Aug 19 2018

MATHEMATICA

Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)

CROSSREFS

Cf. A000009, A294467, A293467, A294468.

Cf. A218481, A294466, A281425, A095051, A294500.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Dec 25 2015

STATUS

approved

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