A375253 Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).

1, 5, 30, 140, 630, 2646, 10710, 41910, 159885, 597025, 2190188, 7914270, 28230020, 99567300, 347720040, 1203777072, 4135047615, 14105322315, 47813634330, 161154659820, 540353553894, 1803226621350, 5991410183850, 19827295283250, 65371101643575

Offset: 0

Author

Seiichi Manyama, Aug 07 2024

Keywords

• nonn

Crossrefs

Column k=4 of A091869 (with a different offset).

Cf. A374506, A375248.

Cf. A005717, A373651.

Programs

Mathematica

a[n_]:=(1+n)(2+n)(3+n)(4+n)Hypergeometric2F1[(1-n)/2,-n/2,2,4]/24; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec((1-2*x+2*x^2)/(1-2*x-3*x^2)^(7/2))

Formula

a(n) = (binomial(n+4,3)/4) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).

a(n) = (binomial(n+4,3)/4) * A005717(n+1).

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

a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([(1-n)/2, -n/2], [2], 4)/24. - Stefano Spezia, Aug 07 2024