A025755 10th-order Patalan numbers (generalization of Catalan numbers).

1, 1, 45, 2850, 206625, 16116750, 1316201250, 110936962500, 9568313015625, 839885253593750, 74749787569843750, 6727480881285937500, 611079513383472656250, 55937278532794804687500

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.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

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.: (11-(1-100*x)^(1/10))/10.

a(n) = 10^(n-1)*9*A035278(n-1)/n!, n >= 2; 9*A035278(n-1) = (10*n-11)(!^10) = Product_{j=2..n} (10*j - 11). - Wolfdieter Lang

Conjecture: n*a(n) + 10*(-10*n+11)*a(n-1) = 0. - R. J. Mathar, Jul 28 2014

a(n) = 100^(n-1)*Pochhammer(9/10, n-1)/n! for n >= 1. Maple confirms this satisfies Mathar's conjecture for n >= 2 (it's not true for n=1). - Robert Israel, Oct 05 2014

Mathematica

CoefficientList[Series[(11 -(1 - 100*x)^(1/10))/10, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)

Crossrefs

Sequence in context: A035097 A184286 A130017 * A213880 A113630 A143004

Adjacent sequences: A025752 A025753 A025754 * A025756 A025757 A025758

Keyword

nonn

Author

Olivier Gérard

Status

approved