A228338 Third diagonal of Catalan difference table (A059346).

5, 9, 19, 43, 102, 250, 628, 1608, 4181, 11009, 29295, 78655, 212815, 579675, 1588245, 4374285, 12103407, 33628827, 93786969, 262450881, 736710360, 2073834420, 5853011850, 16558618510, 46949351275, 133390812255, 379708642289, 1082797114049, 3092894319078, 8848275403642

Offset: 0

Author

N. J. A. Sloane, Aug 29 2013

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Crossrefs

Cf. A059346, A001006, A005043, A005554.

Programs

Maple

a := n -> 5*(-1)^n*hypergeom([7/2, -n], [5], 4):
seq(simplify(a(n)), n=0..29); # Peter Luschny, May 25 2021

Mathematica

CoefficientList[Series[-(x+1)^(5/2)*Sqrt[1-3*x]/(2*x^4)-1/2*(- 1 - x + 3*x^2 + 7*x^3)/x^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 14 2014 *)

PARI

x='x+O('x^50); Vec(-(x+1)^(5/2)*sqrt(1-3*x)/(2*x^4)-1/2*(-1-x+3*x^2+7*x^3)/x^4) \\ G. C. Greubel, May 31 2017

Formula

From Vaclav Kotesovec, Feb 14 2014: (Start)

Recurrence: (n+4)*a(n) = (2*n+7)*a(n-1) + 3*(n-1)*a(n-2).

G.f.: -(x+1)^(5/2)*sqrt(1-3*x)/(2*x^4)-1/2*(-1-x+3*x^2+7*x^3)/x^4.

a(n) ~ 8 * 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). (End)

a(n) = 5*(-1)^n*hypergeom([7/2, -n], [5], 4). - Peter Luschny, May 25 2021

Extensions

Terms a(21) onward added by G. C. Greubel, May 31 2017