%I #37 Jan 05 2025 19:51:38

%S 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,

%T 10,240,1008,960,160,11,330,1848,2640,880,32,12,440,3168,6336,3520,

%U 384,13,572,5148,13728,11440,2496,64,14,728,8008,27456,32032,11648,896,15,910,12012,51480,80080,43680,6720,128

%N T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.

%C Row n contains ceiling(n/2) terms. Row sums yield the Pell numbers (A000129). Column 1 yields A007290.

%C 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

%C 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

%C 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

%H G. C. Greubel, <a href="/A105070/b105070.txt">Rows n = 1..100 of the triangle, flattened</a>

%H Rui Duarte and António Guedes de Oliveira, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Duarte/duarte7.html">A Famous Identity of Hajós in Terms of Sets</a>, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.

%H J. Ivie, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/8-1/elementary8-1.pdf">Problem B-161</a>, Fibonacci Quarterly, 8 (1970), 107-108.

%F E.g.f.: exp(x)*sinh(x*sqrt(2*y))/sqrt(2*y), cf. A034867. - _Vladeta Jovovic_, Apr 06 2005

%F From _Philippe Deléham_, Apr 07 2012: (Start)

%F As DELTA-triangle T(n,k) with 0 <= k <= n:

%F G.f.: (1-x+x^2-y*x^2)/(1-2*x+x^2-2*y*x^2).

%F 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)

%F 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

%e Triangle begins:

%e 1;

%e 2;

%e 3, 2;

%e 4, 8;

%e 5, 20, 4;

%e 6, 40, 24;

%e (2, -1/2, 1/2, 0, 0, ...) DELTA (0, 1, -1, 0, 0, ...) begins:

%e 1;

%e 2, 0;

%e 3, 2, 0;

%e 4, 8, 0, 0;

%e 5, 20, 4, 0, 0;

%e 6, 40, 24, 0, 0, 0.

%e (1, 1, -1, 1, 0, 0, ...) DELTA (0, 0, 2, -2, 0, 0, ...) begins:

%e 1;

%e 1, 0;

%e 2, 0, 0;

%e 3, 2, 0, 0;

%e 4, 8, 0, 0, 0;

%e 5, 20, 4, 0, 0, 0;

%e 6, 40, 24, 0, 0, 0, 0. - _Philippe Deléham_, Apr 07 2012

%p 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

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]

%t v[n_, x_] := u[n - 1, x] + v[n - 1, x]

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A207536 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A105070 *)

%t (* _Clark Kimberling_, Feb 18 2010 *)

%t Table[2^k*Binomial[n, 2*k+1], {n, 15}, {k,0,Floor[(n-1)/2]}]//Flatten (* _G. C. Greubel_, Mar 15 2020 *)

%o (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

%o (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

%Y Cf. A000129, A007290.

%Y Cf. A118397 (eigenvector).

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Apr 05 2005