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URL: https://oeis.org/A115197

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A115197
Convolution of generalized Catalan numbers A064062 (called C(n;2)).
2
1, 2, 7, 32, 169, 974, 5947, 37820, 247885, 1662890, 11362399, 78806936, 553386097, 3926523782, 28108587139, 202764451700, 1472446595221, 10755543924578, 78973277044903, 582558618222416, 4315238786662585
OFFSET
0,2
COMMENTS
Row sums of triangle A115193, called C(1,2).
The o.g.f. given below follows from the Riordan matrix structure of the triangle A115193. See the o.g.f. for the row polynomials of A115193.
LINKS
Joseph Abate and Ward Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, after theorem 5.
Owen John Levens, Joel Brewster Lewis, and Bridget Eileen Tenner, Global patterns in signed permutations, arXiv:2504.13108 [math.CO], 2025. See p. 19.
FORMULA
a(n)= sum(A115193(n,m),m=0..n), n>=0.
G.f.: ((1+2*x*c(2*x))/(1+x))^2 = ((1-2*x) + 6*x*c(2*x))/(1+x)^2, with the o.g.f. c(x) of Catalan numbers A000108.
a(n)= sum(C(2;n-k)*C(2;k),k=0..n), n>=0, with C(2;n):= A064062(n).
a(n)=4*A178792(n)-3*(n+1)*A064062(n+1) [From Joseph Abate, Jun 21 2010]
n*a(n) +(-7*n+13)*a(n-1) +4*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 09 2017
CROSSREFS
Sequence in context: A226269 A006781 A330075 * A107593 A125277 A321688
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Feb 23 2006
STATUS
approved