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

0,3

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.

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

Mathematica

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

Crossrefs

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

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

Sequence in context: A122010 A180911 A084774 * A243668 A265033 A226270

Adjacent sequences: A025754 A025755 A025756 * A025758 A025759 A025760

Keyword

nonn

Author

Olivier Gérard

Status

approved