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a(n) = 5^n *n! *L_n^{-1/5}(-1), where L_n^(alpha)(x) are generalized Laguerre polynomials.
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%I #17 Sep 08 2022 08:45:12

%S 1,9,151,3569,107481,3910349,166221991,8067313749,439513616881,

%T 26531702453969,1756401581518551,126445101373298009,

%U 9830454366312366601,820512954392820339669,73156610496742582153831,6937257394297322500319549,697004737436352946597046241

%N a(n) = 5^n *n! *L_n^{-1/5}(-1), where L_n^(alpha)(x) are generalized Laguerre polynomials.

%H G. C. Greubel, <a href="/A089916/b089916.txt">Table of n, a(n) for n = 0..349</a>

%F a(n) ~ n^(n+3/20)*5^n*exp(-n+2*sqrt(n)-1/2)/sqrt(2) * (1 + 643/(1200*sqrt(n))). - _Vaclav Kotesovec_, Jun 24 2013

%F From _G. C. Greubel_, May 13 2018: (Start)

%F E.g.f.: 1/(1 - 5*x)^(4/5)*exp(5*x/(1 - 5*x)).

%F a(n) = (10*n -1)*a(n-1) - (n-1)*(25*n - 30)*a(n-2). (End)

%p A089916 := proc(n)

%p 5^n*n!*LaguerreL(n,-1/5,-1) ;

%p simplify(%) ;

%p end proc:

%p seq(A089916(n),n=0..10) ; # _R. J. Mathar_, Nov 12 2011

%t Table[5^n*n!*LaguerreL[n,-1/5,-1],{n,0,20}] (* _Vaclav Kotesovec_, Jun 24 2013 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1 - 5*x)^(4/5)*exp(5*x/(1 - 5*x)))) \\ _G. C. Greubel_, May 13 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients( R!(1/(1 - 5*x)^(4/5)*Exp(5*x/(1 - 5*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 13 2018

%K nonn

%O 0,2

%A _Karol A. Penson_, Nov 14 2003

%E Terms a(13) onward added by _G. C. Greubel_, May 13 2018