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A014333
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Three-fold exponential convolution of Catalan numbers with themselves.
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3
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1, 3, 12, 57, 306, 1806, 11508, 78147, 559962, 4201038, 32792472, 264946446, 2206077804, 18860908644, 165050642736, 1474389557739, 13413397423482, 124030117316238, 1163661348170328, 11060842687616610, 106377560784576612, 1034009073326130876
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listen;
history;
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internal format)
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OFFSET
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0,2
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..930
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FORMULA
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E.g.f.: exp(6*x)*(BesselI(0,2*x) - BesselI(1,2*x))^3. - Ilya Gutkovskiy, Nov 01 2017
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: (n+1)*(n+2)*(n+3)*a(n) = 4*(6*n^3 + 13*n^2 + 2*n - 3)*a(n-1) - 4*(n-1)*(44*n^2 - 16*n - 21)*a(n-2) + 192*(n-2)*(n-1)*(2*n - 3)*a(n-3).
a(n) ~ 2^(2*n) * 3^(n + 9/2) / (Pi^(3/2) * n^(9/2)). (End)
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MATHEMATICA
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nmax = 20; CoefficientList[Series[E^(6*x)*(BesselI[0, 2*x] - BesselI[1, 2*x])^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 13 2017 *)
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PROG
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(Magma)
m:=40;
R<x>:=PowerSeriesRing(Rationals(), m);
f:= func< x | (&+[(k+1-x)*x^(2*k)/(Factorial(k)*Factorial(k+1)): k in [0..m+2]]) >;
Coefficients(R!(Laplace( Exp(6*x)*( f(x) )^3 ))); // G. C. Greubel, Jan 06 2023
(SageMath)
m=40
def f(x): return sum((k+1-x)*x^(2*k)/(factorial(k)*factorial(k+1)) for k in range(m+2))
def A014333_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()
A014333_list(m) # G. C. Greubel, Jan 06 2023
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CROSSREFS
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Cf. A000108, A014330, A126869, A138364.
Sequence in context: A159609 A128326 A323631 * A185618 A027710 A307495
Adjacent sequences: A014330 A014331 A014332 * A014334 A014335 A014336
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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