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A068765
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Generalized Catalan numbers.
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1
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1, 1, 6, 39, 270, 1962, 14796, 114831, 911574, 7368894, 60457428, 502162902, 4214515212, 35686162548, 304491863448, 2615468845311, 22598114065254, 196269877811574, 1712578870493316, 15005719955119698
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OFFSET
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0,3
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COMMENTS
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a(n)=K(3,3; n)/3 with K(a,b; n) defined in a comment to A068763.
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LINKS
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FORMULA
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a(n)=(3^n)*p(n, -2/3) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 3*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-12*x*(1-2*x)))/(6*x).
Recurrence: (n+1)*a(n) = 24*(2-n)*a(n-2) + 6*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6+6*sqrt(3)) * (6+2*sqrt(3))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[1-12*x*(1-2*x)])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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