A377216 Expansion of 1/(1 - 4*x^3/(1-x))^(5/2).

1, 0, 0, 10, 10, 10, 80, 150, 220, 710, 1620, 2950, 7010, 16110, 32560, 70682, 156810, 329290, 698540, 1507110, 3189742, 6725150, 14279520, 30141730, 63335960, 133297362, 279996460, 586364410, 1227337710, 2566307410, 5355970048, 11166535430, 23259949980, 48389451510

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, A377215.

Cf. A360309, A377213.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^3/(1-x))^(5/2))); // Vincenzo Librandi, May 10 2025

Mathematica

Table[Sum[(-4)^k*Binomial[-5/2,k]*Binomial[n-2*k-1,n-3*k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 10 2025 *)

PARI

a(n) = sum(k=0, n\3, (-4)^k*binomial(-5/2, k)*binomial(n-2*k-1, n-3*k));

Formula

a(n) = (2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n+9)*a(n-3) - 2*(2*n+2)*a(n-4))/n for n > 3.

a(n) = Sum_{k=0..floor(n/3)} (-4)^k * binomial(-5/2,k) * binomial(n-2*k-1,n-3*k).

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