A089914 - OEIS

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

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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