A382536 Expansion of 1/(1 - x*(1 + 4*x)^(3/2)).

1, 1, 7, 19, 63, 221, 679, 2365, 7499, 25351, 82043, 274031, 892263, 2972127, 9686899, 32261819, 105124711, 350277365, 1140610399, 3803874525, 12372800403, 41319077557, 134176480535, 448958154449, 1454582791283, 4879992151217, 15762304059447, 53067612190093

Offset: 0

Author

Seiichi Manyama, Mar 31 2025

Keywords

• sign
• easy

Comments

a(62) is negative.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A362154, A382537, A382538.

Cf. A002421, A382514.

Programs

Magma

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + 4*x)^(3/2)); seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 01 2025

Mathematica

Table[Sum[4^(n-k)*Binomial[3*k/2,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, Apr 01 2025 *)

PARI

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

Formula

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

D-finite with recurrence (-n+1)*a(n) +2*(-2*n+7)*a(n-1) +(n-1)*a(n-2) +2*(8*n-13)*a(n-3) +24*(4*n-9)*a(n-4) +32*(8*n-23)*a(n-5) +128*(2*n-7)*a(n-6)=0. - R. J. Mathar, Apr 02 2025

a(n) ~ 3 * (-1)^(n+1) * 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Apr 13 2025