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
Keywords
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
Crossrefs
Programs
-
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 *) -
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 */
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