A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

1, 1, 7, 69, 783, 9597, 123495, 1643397, 22413183, 311466829, 4392857431, 62702224213, 903886452975, 13138698859677, 192337495360071, 2832859169364261, 41946319269028191, 624009420903043821

Offset: 0

Author

Olivier Gérard

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• W. 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
• T. M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.

Crossrefs

a(n), n >= 1, = row sums of triangle A049213.

Cf. A000108, A025756, A025758, A025759, A025760, A025761, A025762, A025763.

Programs

Mathematica

Table[SeriesCoefficient[4/(3 + (1 - 16*x)^(1/4)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)

Formula

G.f.: 4 / (3+(1-16*x)^(1/4)).

a(n) = Sum_{m=1..n-1} (m/n*4^(n-m)) * Sum_{k=1..n-m} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j) + 1. - Vladimir Kruchinin, Feb 08 2011

Conjecture: 5*n*(n-1)*(n-2)*a(n) -(239*n-600)*(n-1)*(n-2)*a(n-1) +24*(n-2)*(158*n^2-953*n+1445)*a(n-2) +16*(-1232*n^3+13056*n^2-45949*n+53730)*a(n-3) -128*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jul 28 2014

a(n) = (-1)^(n+1) * 4^(2*n+1) * Sum_{k>=0} (-1/3)^(k+1) * binomial(k/4,n). - Seiichi Manyama, Aug 04 2024