A377204 Expansion of 1/(1 - 4*x^2/(1-x))^(3/2).

1, 0, 6, 6, 36, 66, 236, 546, 1626, 4106, 11388, 29646, 79838, 209718, 557328, 1465970, 3869448, 10166370, 26726080, 70092570, 183756378, 481048010, 1258494768, 3289100958, 8590288128, 22418099982, 58467588768, 152388145382, 396954437202, 1033452111702, 2689186662552

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Formula

a(n) = (2*(n-1)*a(n-1) + (3*n+6)*a(n-2) - 2*(2*n-3)*a(n-3))/n for n > 2.

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

a(n) ~ sqrt(n) * 2^(3*n - 1/2) / (17^(3/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 3/2)). - Vaclav Kotesovec, May 03 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A377197, A377213.

Cf. A026585, A377215.

Cf. A377186.

Sequence in context: A056452 A183622 A176861 * A245131 A038260 A242087

Adjacent sequences: A377201 A377202 A377203 * A377205 A377206 A377207

Keyword

nonn

Author

Seiichi Manyama, Oct 20 2024

Status

approved