A361812 Expansion of 1/sqrt(1 - 4*x*(1+x)^3).

1, 2, 12, 62, 342, 1932, 11094, 64480, 378150, 2233304, 13263772, 79136844, 473969586, 2847911596, 17159547804, 103640073972, 627280131594, 3803643145596, 23102172930156, 140522319418164, 855880464524472, 5219168576004184, 31861229045809436

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(3*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) + (2*n-4)*a(n-4) ) for n > 3.

a(n) = binomial(2*n, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 1/2-n, 2/3-n], -2^6/3^3). - Stefano Spezia, Jul 11 2024

Mathematica

a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a, 23, 0] (* Stefano Spezia, Jul 11 2024 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x)^3))

Crossrefs

Column k=3 of A361830.

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361813, A361814.

Sequence in context: A289787 A226506 A026076 * A231025 A330387 A368760

Adjacent sequences: A361809 A361810 A361811 * A361813 A361814 A361815

Keyword

nonn,changed

Author

Seiichi Manyama, Mar 25 2023

Status

approved