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A228338
Third diagonal of Catalan difference table (A059346).
1
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
0,1
LINKS
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 29 2013
EXTENSIONS
Terms a(21) onward added by G. C. Greubel, May 31 2017
STATUS
approved