A382648 Expansion of 1/(1 - x*(1 + 4*x)^(1/2))^3.

1, 3, 12, 28, 87, 171, 522, 810, 2985, 2583, 18528, -5244, 141875, -241815, 1393314, -3905782, 16326069, -54884079, 209607744, -752322624, 2812050471, -10351091321, 38636724474, -143916146094, 539225694641, -2023036045635, 7615213571172, -28722320569796, 108591659035131

Offset: 0

Author

Seiichi Manyama, Apr 02 2025

Keywords

• sign
• easy

Links

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

Crossrefs

Cf. A382540, A382650.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 28); Coefficients(R!( 1/(1 - x*(1 + 4*x)^(1/2))^3)); // Vincenzo Librandi, May 13 2025

Mathematica

Table[Sum[4^(n-k)* Binomial[k+2,2]* Binomial[k/2, n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 13 2025 *)

PARI

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

Formula

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