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A014333 Three-fold exponential convolution of Catalan numbers with themselves.

%I A014333 #17 Jan 06 2023 15:42:16
%S A014333 1,3,12,57,306,1806,11508,78147,559962,4201038,32792472,264946446,
%T A014333 2206077804,18860908644,165050642736,1474389557739,13413397423482,
%U A014333 124030117316238,1163661348170328,11060842687616610,106377560784576612,1034009073326130876
%N A014333 Three-fold exponential convolution of Catalan numbers with themselves.
%H A014333 G. C. Greubel, <a href="/A014333/b014333.txt">Table of n, a(n) for n = 0..930</a>
%F A014333 E.g.f.: exp(6*x)*(BesselI(0,2*x) - BesselI(1,2*x))^3. - _Ilya Gutkovskiy_, Nov 01 2017
%F A014333 From _Vaclav Kotesovec_, Nov 13 2017: (Start)
%F A014333 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).
%F A014333 a(n) ~ 2^(2*n) * 3^(n + 9/2) / (Pi^(3/2) * n^(9/2)). (End)
%t A014333 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 *)
%o A014333 (Magma)
%o A014333 m:=40;
%o A014333 R<x>:=PowerSeriesRing(Rationals(), m);
%o A014333 f:= func< x | (&+[(k+1-x)*x^(2*k)/(Factorial(k)*Factorial(k+1)): k in [0..m+2]]) >;
%o A014333 Coefficients(R!(Laplace( Exp(6*x)*( f(x) )^3 ))); // _G. C. Greubel_, Jan 06 2023
%o A014333 (SageMath)
%o A014333 m=40
%o A014333 def f(x): return sum((k+1-x)*x^(2*k)/(factorial(k)*factorial(k+1)) for k in range(m+2))
%o A014333 def A014333_list(prec):
%o A014333     P.<x> = PowerSeriesRing(QQ, prec)
%o A014333     return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()
%o A014333 A014333_list(m) # _G. C. Greubel_, Jan 06 2023
%Y A014333 Cf. A000108, A014330, A126869, A138364.
%K A014333 nonn
%O A014333 0,2
%A A014333 _N. J. A. Sloane_