A377199 Expansion of 1/(1 - 4*x/(1-x))^(5/2).

1, 10, 80, 570, 3790, 24062, 147780, 885190, 5199560, 30065870, 171623328, 969151710, 5422863630, 30105497970, 165993714540, 909770119914, 4959840748350, 26912374137150, 145411035749600, 782681600883950, 4198276264607290, 22448626776903450, 119690255236279100

Offset: 0

Author

Seiichi Manyama, Oct 19 2024

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A085362, A377197, A377200.

Programs

Magma

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

Mathematica

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

PARI

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

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * a(k).

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

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

a(n) ~ 128 * n^(3/2) * 5^(n - 5/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Oct 26 2024

a(n) = 10*hypergeom([7/2, 1-n], [2], -4) for n > 0. - Stefano Spezia, May 08 2025