A387281 Expansion of 1/((1-2*x) * (1-6*x))^(7/2).

1, 28, 462, 5880, 63966, 626472, 5692764, 48919728, 402648246, 3202791592, 24780247492, 187393703952, 1390208264172, 10146829592592, 73029572999352, 519260074512480, 3652939914500646, 25457292175929768, 175932472247239092, 1206772898939860560, 8221969006750158660

Offset: 0

Author

Seiichi Manyama, Aug 24 2025

Keywords

• nonn
• new

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A387273.

Programs

Mathematica

Module[{x}, CoefficientList[Series[1/((3*x - 2)*4*x + 1)^(7/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

PARI

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

Formula

n*a(n) = (8*n+20)*a(n-1) - 12*(n+5)*a(n-2) for n > 1.

a(n) = (-2)^n * Sum_{k=0..n} 3^k * binomial(-7/2,k) * binomial(-7/2,n-k).

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

a(n) = Sum_{k=0..n} 4^k * 6^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).

a(n) = (binomial(n+6,3)/20) * A387273(n).

a(n) = (-1)^n * Sum_{k=0..n} 8^k * (3/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).