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

0,2

Links

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A000108, A014330, A126869, A138364.

Sequence in context: A159609 A128326 A323631 * A185618 A027710 A307495

Adjacent sequences: A014330 A014331 A014332 * A014334 A014335 A014336

Keyword

nonn

Author

N. J. A. Sloane

Status

approved