A387284 Expansion of 1/((1-3*x) * (1-7*x))^(7/2).

1, 35, 714, 11130, 147126, 1739430, 18977574, 194933310, 1910889981, 18049106075, 165420791536, 1478846110560, 12948121954404, 111381159472380, 943685046652536, 7890986412386280, 65229211294855839, 533768263527907485, 4328710275989203566, 34823868658927020750

Offset: 0

Author

Seiichi Manyama, Aug 24 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A387238, A387276.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 26 2025

Mathematica

CoefficientList[Series[1/((1-3x)*(1-7*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 26 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(7/2))

Formula

n*a(n) = (10*n+25)*a(n-1) - 21*(n+5)*a(n-2) for n > 1.

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

a(n) = Sum_{k=0..n} (-4)^k * 3^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).

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

a(n) = (binomial(n+6,3)/20) * A387276(n).

a(n) = (-1)^n * Sum_{k=0..n} 10^k * (21/10)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).