A089914 a(n) = 3^n *n! *L_{n}^{-1/3}(-1), where L_n^{alpha}(x) are generalized Laguerre polynomials.

1, 5, 49, 683, 12181, 263093, 6650245, 192153587, 6238115689, 224551351493, 8869372524409, 381149538287675, 17695559832649021, 882309688871504117, 47006884504348992589, 2664275436650330250947, 160032535163170368513745, 10152680666397062173751813, 678253866638401212881394241

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

Formula

a(n) ~ n^(n+1/12)*3^n*exp(-n+2*sqrt(n)-1/2)/sqrt(2) * (1 + 65/(144*sqrt(n))). - Vaclav Kotesovec, Jun 22 2013

From Peter Bala, Jun 14 2014: (Start)

E.g.f.: 1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)) = 1 + 5*x + 49*x^2/2! + 683*x^3/3! + ....

Dobinski-type formula: a(n) = (3^n/exp(1))*Sum {k >= 0} (n!/k!)* binomial(n + k - 1/3,k - 1/3).

Recurrence equation: a(n) = (6*n - 1)a(n-1) - (n - 1)*(9*n - 12)*a(n-2) with a(0) = 1 and a(1) = 5. (End)

Maple

A089914 := proc(n)
        3^n*n!*LaguerreL(n, -1/3, -1) ;
        simplify(%) ;
end proc;

Mathematica

Table[3^n*n!*LaguerreL[n, -1/3, -1], {n, 0, 20}] (* Vaclav Kotesovec, Jun 22 2013 *)

Prog

(PARI) x='x+O('x^30); Vec(serlaplace(1/(1 - 3*x)^(2/3)*exp(3*x/(1 - 3*x)))) \\ G. C. Greubel, May 13 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients( R!(1/(1 - 3*x)^(2/3)*Exp(3*x/(1 - 3*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 13 2018

Crossrefs

Sequence in context: A228511 A116873 A324361 * A267220 A052142 A136729

Adjacent sequences: A089911 A089912 A089913 * A089915 A089916 A089917

Keyword

nonn

Author

Karol A. Penson, Nov 14 2003

Extensions

Terms a(15) onward added by G. C. Greubel, May 13 2018

Status

approved