A025756 3rd-order Vatalan numbers (generalization of Catalan numbers).

1, 1, 4, 22, 139, 949, 6808, 50548, 384916, 2988418, 23559826, 188061592, 1516680130, 12337999870, 101111413540, 833914857316, 6916004156083, 57638242134229, 482444724374734, 4053815358183454, 34181335453533439

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880, 2014 and J. Int. Seq. 18 (2015) # 15.3.3

Formula

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

a(n) = Sum_{m=1..n-1} (m/n) * Sum_{k=1..n-m} binomial(k,n-m-k) * 3^k * (-1)^(n-m-k) * binomial(n+k-1,n-1) + 1. - Vladimir Kruchinin, Feb 08 2011

Conjecture: n*(n-1)*a(n) -(n-1)*(19*n-36)*a(n-1) +9*(11*n^2-51*n+60)*a(n-2) -9*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011

a(n) ~ 9^n/(4*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Oct 08 2012

a(n) = (-1)^(n+1) * 3^(2*n+1) * Sum_{k>=0} (-1/2)^(k+1) * binomial(k/3,n). - Seiichi Manyama, Aug 04 2024

Maple

A025756 := proc(n)
    coeftayl( 3/(2+(1-9*x)^(1/3)), x=0, n);
end proc:
seq(A025756(n), n=0..30); # Wesley Ivan Hurt, Aug 02 2014

Mathematica

Table[SeriesCoefficient[3/(2+(1-9*x)^(1/3)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)

Prog

(Maxima) a[0]:1$ a[n]:=(1/n)*((9*n-6)*a[n-1]-2*sum(a[k]*a[n-1-k], k, 0, n-1))$ makelist(a[n], n, 0, 1000); /* Tani Akinari, Aug 02 2014 */

Crossrefs

Row sums of triangle A048966, n > 0.

Cf. A000108, A025757, A025758, A025759, A025760, A025761, A025762, A025763.

Sequence in context: A283055 A097593 A188686 * A366119 A200731 A193116

Adjacent sequences: A025753 A025754 A025755 * A025757 A025758 A025759

Keyword

nonn

Author

Olivier Gérard

Status

approved