A374497 Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).

1, 6, 36, 200, 1080, 5712, 29792, 153792, 787680, 4009280, 20304768, 102405888, 514678528, 2579028480, 12890311680, 64283809792, 319954540032, 1589720712192, 7886437652480, 39069462835200, 193307835764736, 955361266917376, 4716674314223616, 23264437702656000

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

a(0) = 1, a(1) = 6; a(n) = (2*(2*n+1)*a(n-1) + 4*(n+1)*a(n-2))/n.

a(n) = binomial(n+2,2) * A071356(n).

a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 20 2025

Mathematica

a[n_]:= Sum[(2*k+1)*Binomial[2*k, k]*Binomial[k, n-k], {k, 0, n}]; Array[a, 24, 0] (* Stefano Spezia, May 08 2025 *)

Prog

(PARI) a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

Crossrefs

Cf. A001788, A002457, A025163.

Cf. A006139, A374511, A374513.

Cf. A071356.

Sequence in context: A200782 A055299 A261520 * A345340 A232138 A000551

Adjacent sequences: A374494 A374495 A374496 * A374498 A374499 A374500

Keyword

nonn

Author

Seiichi Manyama, Jul 09 2024

Status

approved