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A025756 3rd order Vatalan numbers (generalization of Catalan numbers). 2
1, 1, 4, 22, 139, 949, 6808, 50548, 384916, 2988418, 23559826, 188061592, 1516680130, 12337999870, 101111413540, 833914857316, 6916004156083, 57638242134229, 482444724374734, 4053815358183454, 34181335453533439 (list; graph; refs; listen; history; text; internal format)
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

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.

Sequence in context: A283055 A097593 A188686 * A200731 A193116 A187254

Adjacent sequences:  A025753 A025754 A025755 * A025757 A025758 A025759

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified February 27 11:38 EST 2021. Contains 341656 sequences. (Running on oeis4.)