A155579 Recursive sequence (n+1)*a(n) = 3*(3*n-2)*a(n-1).

2, 3, 12, 63, 378, 2457, 16848, 120042, 880308, 6602310, 50417640, 390736710, 3065780340, 24307258410, 194458067280, 1567818167445, 12726994535730, 103937122041795, 853378475711580, 7040372424620535, 58334514375427290

Offset: 0

Author

Roger L. Bagula, Jan 24 2009

Keywords

• nonn
• easy

Comments

This is built akin to (n+1)*C(n) = 2*(2*n-1)*C(n-1) for the Catalan numbers A000108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A185047, A283150.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = ((3*n - 2)/(n + 1))*a[n - 1];
Table[2*3^(n)*a[n], {n, 0, 30}]

Maxima

a(n):=3^(2*n-1)*sum(binomial(k,n-k)*2^(2*k-n)*(-1)^(n-k)*(if k=1 then (1/3) else 1/k*(1/3)^k*sum(binomial(i,k-1-i)*(-1/3)^(k-1-i)*binomial(i+k-1,k-1),i,1,k-1)),k,1,n); /* Vladimir Kruchinin, Sep 20 2010 */

Formula

(n+1)*a(n) = 3*(3*n-2)*a(n-1).

From Vladimir Kruchinin, Sep 20 2010: (Start)

G.f.: A(x) = 1/3*(1-(1-9*x)^(2/3)).

a(n) = 3^(2*n-1)*sum(binomial(k,n-k)*2^(2*k-n)*(-1)^(n-k)*(if k=1 then (1/3) else 1/k*(1/3)^k*sum(binomial(i,k-1-i)*(-1/3)^(k-1-i)*binomial(i+k-1,k-1),i,1,k-1)),k,1,n),n>0. (End)

From Vaclav Kotesovec, Jul 20 2019: (Start)

a(n) = 2 * 3^(2*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n+2)).

a(n) ~ 2 * 3^(2*n) / (Gamma(1/3) * n^(5/3)). (End)

a(n) = 3*A185047(n-1) for n >= 1. - Peter Bala, Oct 14 2024