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A137572
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The first upper diagonal of square array A137570; equals the convolution of the main diagonal A137571 with A002293.
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3
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1, 3, 16, 100, 681, 4908, 36842, 285158, 2260257, 18257902, 149769225, 1244277499, 10448404901, 88538107802, 756153001241, 6501989278168, 56244305146039, 489111092027854, 4273491476147117, 37496699100314116, 330261353255659842
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| G.f. A(x) = F(x)/(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 + 3*x + 16*x^2 + 100*x^3 + 681*x^4 + 4908*x^5 +...;
A(x) = F(x)/(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=F/(1-x*C*F^2-x*F^3); polcoeff(A+O(x^m), n, x)}
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CROSSREFS
| Cf. A137570, A137571, A137573; A000108, A002293.
Sequence in context: A074553 A193037 A091641 * A009151 A009007 A000949
Adjacent sequences: A137569 A137570 A137571 * A137573 A137574 A137575
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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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