A139548 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, 2, 4, 2, 4, 16, 24, 16, 4, 8, 48, 120, 160, 120, 48, 8, 16, 128, 448, 896, 1120, 896, 448, 128, 16, 32, 320, 1440, 3840, 6720, 8064, 6720, 3840, 1440, 320, 32, 64, 768, 4224, 14080, 31680, 50688, 59136, 50688, 31680, 14080, 4224, 768, 64, 128, 1792, 11648

Offset

0,2

Comments

Row sums are A001018

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 100)

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Formula

T(n,k) = 2^n*binomial(2*n,k). - R. J. Mathar, Sep 12 2013

Example

1;
2, 4, 2;
4, 16, 24, 16, 4;
8, 48, 120, 160, 120, 48, 8;
16, 128, 448, 896, 1120, 896, 448, 128, 16;
32, 320, 1440, 3840, 6720, 8064, 6720, 3840, 1440, 320, 32;
64, 768, 4224, 14080, 31680, 50688, 59136, 50688, 31680, 14080, 4224, 768, 64;

Mathematica

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]
(* Second program: *)
Table[2^n*Binomial[2 n, k], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* Michael De Vlieger, May 16 2018 *)

Crossrefs

Cf. A001018.

Sequence in context: A241078 A198285 A136620 * A193378 A108445 A263424

Adjacent sequences: A139545 A139546 A139547 * A139549 A139550 A139551

Keyword

nonn,tabf

Author

Roger L. Bagula and Gary W. Adamson, Jun 10 2008

Status

approved