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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. 3
1, 5, 29, 186, 1281, 9294, 70109, 544833, 4333381, 35108351, 288738813, 2404256945, 20228988678, 171716799066, 1468804301441, 12647321103329, 109538312419238, 953622158606749, 8340394595266367, 73247287493299642 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

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.

EXAMPLE

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), ...].

PROG

(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)}

CROSSREFS

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

Sequence in context: A081336 A127846 A059231 * A234317 A078945 A113713

Adjacent sequences:  A137570 A137571 A137572 * A137574 A137575 A137576

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2008

STATUS

approved

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Last modified February 17 18:46 EST 2019. Contains 320222 sequences. (Running on oeis4.)