A115197 Convolution of generalized Catalan numbers A064062 (called C(n;2)).

1, 2, 7, 32, 169, 974, 5947, 37820, 247885, 1662890, 11362399, 78806936, 553386097, 3926523782, 28108587139, 202764451700, 1472446595221, 10755543924578, 78973277044903, 582558618222416, 4315238786662585

Offset: 0

Author

Wolfdieter Lang, Feb 23 2006

Keywords

• nonn

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