A361790 Expansion of 1/sqrt(1 - 4*x/(1+x)^4).

1, 2, -2, -8, 6, 42, -8, -228, -90, 1210, 1238, -6116, -10864, 28574, 80932, -116248, -548010, 339678, 3455686, 173208, -20452674, -14036418, 113365140, 156407916, -580805472, -1312098918, 2659610562, 9621079540, -9902139124, -64566648122, 18521111032

Offset

0,2

Links

Winston de Greef, Table of n, a(n) for n = 0..2497

Formula

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

n*a(n) = -( (n-3)*a(n-1) + (6*n-6)*a(n-2) + 10*(n-3)*a(n-3) + 5*(n-4)*a(n-4) + (n-5)*a(n-5) ) for n > 4.

a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (-1)^(n-1-k) * (n+k) * binomial(n+2-k,3) * a(k).

a(n) = (-1)^(n+1)*Pochhammer(n,3)*hypergeom([1-n, 1+n/3, (4+n)/3, (5+n)/3], [5/4, 7/4, 2], 3^3/2^6)/3 for n > 0. - Stefano Spezia, Jul 11 2024

Mathematica

a[n_]:=(-1)^(n+1)Pochhammer[n, 3]HypergeometricPFQ[{1-n, 1+n/3, (4+n)/3, (5+n)/3}, {5/4, 7/4, 2}, 3^3/2^6]/3; Join[{1}, Array[a, 30]] (* Stefano Spezia, Jul 11 2024 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1+x)^4))
(PARI) a(n)=sum(k=0, n, (-1)^(n-k) * binomial(2*k, k) * binomial(n+3*k-1, n-k)) \\ Winston de Greef, Mar 24 2023

Crossrefs

Cf. A006139, A137635, A360133, A361791, A361792.

Sequence in context: A071418 A353268 A245582 * A197820 A064862 A123202

Adjacent sequences: A361787 A361788 A361789 * A361791 A361792 A361793

Keyword

sign

Author

Seiichi Manyama, Mar 24 2023

Status

approved