login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
1

%I #12 May 12 2020 20:59:16

%S 1,2,4,2,4,16,24,16,4,8,48,120,160,120,48,8,16,128,448,896,1120,896,

%T 448,128,16,32,320,1440,3840,6720,8064,6720,3840,1440,320,32,64,768,

%U 4224,14080,31680,50688,59136,50688,31680,14080,4224,768,64,128,1792,11648

%N Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.

%C Row sums are A001018

%H Michael De Vlieger, <a href="/A139548/b139548.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 100)

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/2002.06672">A bracket polynomial for 2-tangle shadows</a>, arXiv:2002.06672 [math.CO], 2020.

%F T(n,k) = 2^n*binomial(2*n,k). - _R. J. Mathar_, Sep 12 2013

%e 1;

%e 2, 4, 2;

%e 4, 16, 24, 16, 4;

%e 8, 48, 120, 160, 120, 48, 8;

%e 16, 128, 448, 896, 1120, 896, 448, 128, 16;

%e 32, 320, 1440, 3840, 6720, 8064, 6720, 3840, 1440, 320, 32;

%e 64, 768, 4224, 14080, 31680, 50688, 59136, 50688, 31680, 14080, 4224, 768, 64;

%t Clear[f, x, n] f[x_, y_, n_] = Sum[Binomial[n, i]*x^i*y^(n - i), {i, 0, n}]; Table[ExpandAll[f[x, y, n]*f[y, z, n]*f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n]*f[y, z, n]*f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]

%t (* Second program: *)

%t Table[2^n*Binomial[2 n, k], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* _Michael De Vlieger_, May 16 2018 *)

%Y Cf. A001018.

%K nonn,tabf

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 10 2008