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

1, 3, 12, 57, 306, 1806, 11508, 78147, 559962, 4201038, 32792472, 264946446, 2206077804, 18860908644, 165050642736, 1474389557739, 13413397423482, 124030117316238, 1163661348170328, 11060842687616610, 106377560784576612, 1034009073326130876

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..930

Crossrefs

Cf. A000108, A014330, A126869, A138364.

Programs

Magma

m:=40;
R:=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

Mathematica

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 *)

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. = PowerSeriesRing(QQ, prec)
    return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()
A014333_list(m) # G. C. Greubel, Jan 06 2023

Formula

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)