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A137573
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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.
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3
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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; internal format)
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OFFSET
| 0,2
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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.
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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), ...].
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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)}
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CROSSREFS
| Cf. A137570, A137571, A137572; A000108, A002293.
Sequence in context: A081336 A059231 A127846 * A078945 A113713 A142980
Adjacent sequences: A137570 A137571 A137572 * A137574 A137575 A137576
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 27 2008
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