A387342 Expansion of 1/(1 - 6*x + x^2)^(7/2).

1, 21, 280, 3024, 28854, 253638, 2103024, 16689816, 128014887, 955485531, 6974119152, 49965080256, 352366829724, 2451595670748, 16858071545664, 114737706591984, 773866620578205, 5177539121330961, 34391021091689416, 226956883258736400, 1488970185631858930

Offset: 0

Author

Seiichi Manyama, Aug 27 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A387338.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 6*x + x^2)^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

Mathematica

CoefficientList[Series[1/(1-6*x+x^2)^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/(1-6*x+x^2)^(7/2))

Formula

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

a(n) = (binomial(n+6,3)/20) * A387338(n).

a(n) = (-1)^n * Sum_{k=0..n} (1/6)^(n-2*k) * binomial(-7/2,k) * binomial(k,n-k).