A387367 Expansion of 1/(1 - 10*x + x^2)^(5/2).

1, 25, 435, 6475, 88270, 1137150, 14081970, 169370850, 1991916135, 23013193775, 262062237437, 2948690451525, 32845189782860, 362721036253100, 3975956599494420, 43300257350934900, 468875116313950845, 5051523021827188725, 54177811767428268535, 578700942412768257775

Offset: 0

Author

Seiichi Manyama, Aug 27 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000332, A387341.

Cf. A387369.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 10*x + x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 29 2025

Mathematica

CoefficientList[Series[1/(1-10*x+x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 29 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+x^2)^(5/2))

Formula

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

a(n) = (binomial(n+4,2)/6) * A387369(n).

a(n) = (-1)^n * Sum_{k=0..n} (1/10)^(n-2*k) * binomial(-5/2,k) * binomial(k,n-k).