A375248 Expansion of (1 - x)/(1 - 2*x - 3*x^2)^(7/2).

1, 6, 35, 168, 756, 3192, 12936, 50688, 193479, 722722, 2651649, 9581936, 34176324, 120526056, 420852204, 1456709328, 5002984791, 17062825626, 57827993685, 194871361608, 653285629920, 2179701604080, 7241015510820, 23958512912880, 78978801164445

Offset: 0

Author

Seiichi Manyama, Aug 07 2024

Keywords

• nonn

Crossrefs

First differences of A374506.

Cf. A046717, A109188, A371408.

Cf. A014531, A375253.

Programs

Mathematica

a[n_]:=(1+n)(2+n)(3+n)(4+n)(5+n)Hypergeometric2F1[(1-n)/2,-n/2,3,4]/120; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

PARI

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

Formula

a(n) = (binomial(n+5,3)/10) * Sum_{k=0..floor(n/2)} binomial(n+2,n-2*k) * binomial(2*k+2,k).

a(n) = (binomial(n+5,3)/10) * A014531(n+1).

a(n) = ((n+5)/(n*(n+4))) * ((2*n+3)*a(n-1) + 3*(n+4)*a(n-2)).

a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*hypergeom([(1-n)/2, -n/2], [3], 4)/120. - Stefano Spezia, Aug 07 2024