A014333 - OEIS

%I #17 Jan 06 2023 15:42:16

%S 1,3,12,57,306,1806,11508,78147,559962,4201038,32792472,264946446,

%T 2206077804,18860908644,165050642736,1474389557739,13413397423482,

%U 124030117316238,1163661348170328,11060842687616610,106377560784576612,1034009073326130876

%N Three-fold exponential convolution of Catalan numbers with themselves.

%H G. C. Greubel, <a href="/A014333/b014333.txt">Table of n, a(n) for n = 0..930</a>

%F E.g.f.: exp(6*x)*(BesselI(0,2*x) - BesselI(1,2*x))^3. - _Ilya Gutkovskiy_, Nov 01 2017

%F From _Vaclav Kotesovec_, Nov 13 2017: (Start)

%F 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 a(n) ~ 2^(2*n) * 3^(n + 9/2) / (Pi^(3/2) * n^(9/2)). (End)

%t 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 (Magma)

%o m:=40;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o f:= func< x | (&+[(k+1-x)*x^(2*k)/(Factorial(k)*Factorial(k+1)): k in [0..m+2]]) >;

%o Coefficients(R!(Laplace( Exp(6*x)*( f(x) )^3 ))); // _G. C. Greubel_, Jan 06 2023

%o (SageMath)

%o m=40

%o def f(x): return sum((k+1-x)*x^(2*k)/(factorial(k)*factorial(k+1)) for k in range(m+2))

%o def A014333_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()

%o A014333_list(m) # _G. C. Greubel_, Jan 06 2023

%Y Cf. A000108, A014330, A126869, A138364.

%K nonn

%O 0,2

%A _N. J. A. Sloane_