A331323 a(n) = [x^n] (1 - 2*x)/(1 - 8*x + 4*x^2)^(3/2).

1, 10, 90, 772, 6430, 52524, 423220, 3375880, 26720118, 210195100, 1645295212, 12825551160, 99633196780, 771702434104, 5961969066600, 45958506432016, 353585912577190, 2715647948258940, 20824876515839932, 159474192002499160, 1219708190630800836, 9318143974952519080

Offset: 0

Author

Peter Luschny, Jan 18 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column 5 of A331511.

Cf. A331431.

Programs

Magma

[(&+[3^k*(k+1)*Binomial(n+1,k+1)^2: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Mar 22 2022

Maple

gf := (1-2*x)/(4*x^2-8*x+1)^(3/2): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0..21); # Or:
a := proc(n) option remember; if n<3 then [1, 10, 90][n+1] else
(10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n fi end:
seq(a(n), n=0..21);

Mathematica

a[n_] := Sum[3^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jan 20 2020 *)

PARI

{a(n) = 2^n*sum(k=0, n, (n+k+1)*binomial(n, k)*binomial(n+k, k)/2^k)} \\ Seiichi Manyama, Jan 18 2020

PARI

N=20; x='x+O('x^N); Vec((1-2*x)/(4*x^2-8*x+1)^(3/2)) \\ Seiichi Manyama, Jan 18 2020

PARI

{a(n) = sum(k=0, n, 3^k*(k+1)*binomial(n+1, k+1)^2)} \\ Seiichi Manyama, Jan 20 2020

Sage

[sum(2^(n-k)*(n+k+1)*binomial(2*k,k)*binomial(n+k,2*k)  for k in (0..n)) for n in (0..30)] # G. C. Greubel, Mar 22 2022

Formula

a(n) = (-2)^n*Sum_{k=0..n} A331431(n, k)/(-2)^k.

a(n) = (10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n.

a(n) = 2^(n-1)*(n+1)*(n*hypergeom([1-n, n+2], [2], -1/2) + 2*hypergeom([-n, n+1], [1], -1/2)).

a(n) = Sum_{k=0..n} 3^k * (k+1) * binomial(n+1,k+1)^2. - Seiichi Manyama, Jan 20 2020

a(n) = (n + 1)^2*hypergeom([-n, -n], [2], 3). - Peter Luschny, Jan 20 2020

n * (2*n-1) * a(n) = 2 * (8 * n^2 - 3) * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1. - Seiichi Manyama, Jan 25 2020