A360133 Expansion of 1/sqrt(1 - 4*x/(1+x)^3).

1, 2, 0, -4, -4, 6, 18, 4, -48, -70, 60, 288, 170, -686, -1386, 432, 4928, 4806, -9684, -27572, -3672, 84106, 118162, -122388, -537834, -284830, 1386840, 2688944, -1103362, -10181934, -9354198, 21404728, 57921144, 3663942, -185437360, -248708676, 292137656

Offset: 0

Author

Seiichi Manyama, Mar 24 2023

Keywords

• sign

Links

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

Crossrefs

Cf. A006139, A137635, A361790, A361791, A361792.

Programs

Mathematica

a[n_]:=(-1)^(n+1)n(n+1)HypergeometricPFQ[{3/2,1-n,1+n/2,(3+n)/2}, {4/3,5/3,2}, 2^4/3^3]; Join[{1},Array[a,36]] (* Stefano Spezia, Jul 11 2024 *)

PARI

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^3))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(n+2*k-1,n-k).

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

a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+1-k,2) * a(k).

a(n) = (-1)^(n+1)*n*(n + 1)*hypergeom([3/2, 1-n, 1+n/2, (3+n)/2], [4/3, 5/3, 2], 2^4/3^3) for n > 0. - Stefano Spezia, Jul 11 2024