A377215 Expansion of 1/(1 - 4*x^2/(1-x))^(5/2).

1, 0, 10, 10, 80, 150, 640, 1550, 5190, 13870, 41912, 115650, 333490, 925970, 2607540, 7220062, 20053700, 55230870, 152005380, 416295350, 1137980678, 3100453710, 8429823180, 22862244210, 61882724100, 167159512794, 450739897980, 1213298505770, 3260824389510

Offset: 0

Author

Seiichi Manyama, Oct 20 2024

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Crossrefs

Cf. A377199, A377216.

Cf. A026585, A377204.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(5/2))); // Vincenzo Librandi, May 08 2025

Mathematica

Table[Sum[(-4)^k*Binomial[-5/2,k]*Binomial[n-k-1,n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)

PARI

a(n) = sum(k=0, n\2, (-4)^k*binomial(-5/2, k)*binomial(n-k-1, n-2*k));

Formula

a(n) = (2*(n-1)*a(n-1) + (3*n+14)*a(n-2) - 2*(2*n-1)*a(n-3))/n for n > 2.

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-5/2,k) * binomial(n-k-1,n-2*k).

a(n) ~ n^(3/2) * 2^(3*n - 1/2) / (3 * 17^(5/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 5/2)). - Vaclav Kotesovec, May 03 2025