A377213 Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).

1, 0, 0, 6, 6, 6, 36, 66, 96, 266, 576, 1026, 2246, 4866, 9516, 19598, 41286, 83526, 170048, 351378, 716850, 1458098, 2984028, 6087270, 12380900, 25224222, 51356400, 104380510, 212164362, 431148222, 875353220, 1776567762, 3604752672, 7310374010, 14819370480, 30033014994

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1800

Formula

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

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

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

Mathematica

Table[Sum[(2*k+1)*Binomial[2*k, k]*Binomial[n-2*k-1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 35}] (* Vincenzo Librandi, May 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (2*k+1)*binomial(2*k, k)*binomial(n-2*k-1, n-3*k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^3/(1-x))^(3/2))); // Vincenzo Librandi, May 08 2025

Crossrefs

Cf. A377197, A377204.

Cf. A360309, A377216.

Sequence in context: A054435 A019175 A245100 * A024953 A092281 A082796

Adjacent sequences: A377210 A377211 A377212 * A377214 A377215 A377216

Keyword

nonn

Author

Seiichi Manyama, Oct 20 2024

Status

approved