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A105070
T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.
+10
6
1, 2, 3, 2, 4, 8, 5, 20, 4, 6, 40, 24, 7, 70, 84, 8, 8, 112, 224, 64, 9, 168, 504, 288, 16, 10, 240, 1008, 960, 160, 11, 330, 1848, 2640, 880, 32, 12, 440, 3168, 6336, 3520, 384, 13, 572, 5148, 13728, 11440, 2496, 64, 14, 728, 8008, 27456, 32032, 11648, 896, 15, 910, 12012, 51480, 80080, 43680, 6720, 128
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row n contains ceiling(n/2) terms. Row sums yield the Pell numbers (A000129). Column 1 yields A007290.
Eigenvector equals A118397, so that A118397(n) = Sum_{k=0..[n/2]} T(n+1,k)*A118397(k) for n >= 0. - Paul D. Hanna, May 08 2006
Essentially a triangle, read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 07 2012
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
J. Ivie, Problem B-161, Fibonacci Quarterly, 8 (1970), 107-108.
FORMULA
E.g.f.: exp(x)*sinh(x*sqrt(2*y))/sqrt(2*y), cf. A034867. - Vladeta Jovovic, Apr 06 2005
From Philippe Deléham, Apr 07 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x+x^2-y*x^2)/(1-2*x+x^2-2*y*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = T(2,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k<0 or if k>n. (End)
Sum_{k=0..floor((n-1)/2)} T(n,k) = { P(n) (A000129(n)), A215928(n), (-1)^(n-1) *A077985(n-1), -A176981(n+1), (-1)^(n-1)*A215936(n+2) }, for n >= 1. - G. C. Greubel, Mar 15 2020
EXAMPLE
Triangle begins:
1;
2;
3, 2;
4, 8;
5, 20, 4;
6, 40, 24;
(2, -1/2, 1/2, 0, 0, ...) DELTA (0, 1, -1, 0, 0, ...) begins:
1;
2, 0;
3, 2, 0;
4, 8, 0, 0;
5, 20, 4, 0, 0;
6, 40, 24, 0, 0, 0.
(1, 1, -1, 1, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, ...) begins:
1;
1, 0;
2, 0, 0;
3, 2, 0, 0;
4, 8, 0, 0, 0;
5, 20, 4, 0, 0, 0;
6, 40, 24, 0, 0, 0, 0. - Philippe Deléham, Apr 07 2012
MAPLE
T:=(n, k)->binomial(n, 2*k+1)*2^k:for n from 1 to 15 do seq(T(n, k), k=0..floor((n-1)/2)) od; # yields sequence in triangular form
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A207536 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A105070 *)
(* Clark Kimberling, Feb 18 2010 *)
Table[2^k*Binomial[n, 2*k+1], {n, 15}, {k, 0, Floor[(n-1)/2]}]//Flatten (* G. C. Greubel, Mar 15 2020 *)
PROG
(Magma) [2^k*Binomial(n, 2*k+1): k in [0..Floor((n-1)/2)], n in [1..15]]; // G. C. Greubel, Mar 15 2020
(Sage) [[2^k*binomial(n, 2*k+1) for k in (0..floor((n-1)/2))] for n in (1..15)] # G. C. Greubel, Mar 15 2020
CROSSREFS
Cf. A000129, A007290.
Cf. A118397 (eigenvector).
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 05 2005
STATUS
approved
A270863
Self-composition of the Fibonacci sequence.
+10
6
0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0, f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1, f(x) g.f. A000045 and g(x) g.f. A000045
k=2, f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3, f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4, f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5, f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6, f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7, f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).
FORMULA
a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022
EXAMPLE
a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.
MAPLE
f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);
PROG
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, -3, 1, 3]^(n-1)*[1; 2; 6; 17])[1, 1] \\ Charles R Greathouse IV, Mar 24 2016
(PARI) concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016
(Magma) I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019
CROSSREFS
Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.
KEYWORD
nonn,easy
AUTHOR
Oboifeng Dira, Mar 24 2016
STATUS
approved
A147746
Riordan array (1, x(1-2x)/(1-3x+x^2)).
+10
5
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 13, 14, 9, 4, 1, 0, 34, 40, 28, 14, 5, 1, 0, 89, 114, 87, 48, 20, 6, 1, 0, 233, 323, 267, 161, 75, 27, 7, 1, 0, 610, 910, 809, 528, 270, 110, 35, 8, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Triangle [0,1,1,1,0,0,0,....] DELTA [1,0,0,0,...] with Deléham DELTA as in A084938.
Note that 1/(1-x/(1-x/(1-x))) = (1-2x)/(1-3x+x^2). Row sums are A124302.
A147746*A007318 = A147747.
LINKS
Table of n, a(n) for n=0..54.
FORMULA
Sum_{k=0..n} T(n,k)*2^k = A147748(n). - Philippe Deléham, Oct 30 2011
Sum_{k=0..n} T(n,k)*(-1)^(n-k) = A215936(n). - Philippe Deléham, Aug 30 2012
G.f.: (1 - 3*x + x^2)/(1 - 3*x + x^2 - x*y + 2*x^2*y). - R. J. Mathar, Aug 11 2015
EXAMPLE
Triangle begins
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 5, 5, 3, 1;
0, 13, 14, 9, 4, 1;
0, 34, 40, 28, 14, 5, 1;
0, 89, 114, 87, 48, 20, 6, 1;
...
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, # (1-2#)/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Nov 11 2008
STATUS
approved
A188285
Riordan matrix ( (1-2x)/(1-2x-x^2), (x-2x^2)/(1-2x-x^2) ).
+10
1
1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 4, 3, 0, 1, 12, 11, 6, 4, 0, 1, 29, 28, 18, 8, 5, 0, 1, 70, 72, 48, 26, 10, 6, 0, 1, 169, 184, 130, 72, 35, 12, 7, 0, 1, 408, 469, 348, 204, 100, 45, 14, 8, 0, 1, 985, 1192, 927, 568, 295, 132, 56, 16, 9, 0, 1, 2378, 3022, 2456, 1571, 850, 404, 168, 68, 18, 10, 0, 1, 5741, 7644, 6477, 4312, 2430, 1200, 532, 208, 81, 20, 11, 0, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
T(n,k) is the number of Dyck paths of height at most 3 with length 2n and k hills.
Row sum = F_(2n-1) Fibonacci number.
T is the convolution triangle of |A215936|. - Peter Luschny, Oct 19 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..495
FORMULA
T(n,k) = sum(M(i,n-k-2i)*Binomial(i+k,k)*2^{n-k-2i},i=0..floor((n-k)/2)), where M(n,k)=n(n+1)(n+2)...(n+k-1)/k!.
Recurrence: T(n+2,k+1) = 2 T(n+1,k+1) + T(n+1,k) + T(n,k+1) - 2 T(n,k)
EXAMPLE
Triangle begins:
1
0, 1
1, 0, 1
2, 2, 0, 1
5, 4, 3, 0, 1
12, 11, 6, 4, 0, 1
29, 28, 18, 8, 5, 0, 1
70, 72, 48, 26, 10, 6, 0, 1
169, 184, 130, 72, 35, 12, 7, 0, 1
408, 469, 348, 204, 100, 45, 14, 8, 0, 1
MAPLE
# Uses function PMatrix from A357368. Adds column 1, 0, 0, 0, ... to the left.
PMatrix(10, n -> (-1)^(n+1)*A215936(n)); # Peter Luschny, Oct 19 2022
MATHEMATICA
Flatten[Table[Sum[Pochhammer[i, n-k-2i]/(n-k-2i)!Binomial[i+k, k]2^(n-k-2i), {i, 0, (n-k)/2}], {n, 0, 12}, {k, 0, n}], 1]
PROG
(Maxima) create_list(sum(pochhammer(i, n-k-2*i)/(n-k-2*i)!*binomial(i+k, k)*2^(n-k-2*i), i, 0, (n-k)/2), n, 0, 12, k, 0, n);
KEYWORD
nonn,tabl,easy
AUTHOR
Emanuele Munarini, Mar 26 2011
STATUS
approved
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