A137573 The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.

1, 5, 29, 186, 1281, 9294, 70109, 544833, 4333381, 35108351, 288738813, 2404256945, 20228988678, 171716799066, 1468804301441, 12647321103329, 109538312419238, 953622158606749, 8340394595266367, 73247287493299642

Offset: 0

Author

Paul D. Hanna, Jan 27 2008

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 5*x + 29*x^2 + 186*x^3 + 1281*x^4 + 9294*x^5 +...;
A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
		

Crossrefs

Cf. A137570, A137571, A137572; A000108, A002293.

Programs

PARI

{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=C*F^2/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

Formula

G.f. A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.