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A289212 a(n) = n! * Laguerre(n,-6).

%I A289212 #27 Feb 16 2025 08:33:48
%S A289212 1,7,62,654,7944,108696,1649232,27422352,495057024,9631281024,
%T A289212 200682406656,4455296877312,104921038236672,2610989435003904,
%U A289212 68430995893131264,1883330926998829056,54286270223002140672,1635031821385383247872,51347572582353094508544
%N A289212 a(n) = n! * Laguerre(n,-6).
%H A289212 Alois P. Heinz, <a href="/A289212/b289212.txt">Table of n, a(n) for n = 0..434</a>
%H A289212 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289212 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289212 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289212 E.g.f.: exp(6*x/(1-x))/(1-x).
%F A289212 a(n) = n! * Sum_{i=0..n} 6^i/i! * binomial(n,i).
%F A289212 a(n) = n! * A160607(n)/A160608(n).
%F A289212 a(n) ~ exp(-3 + 2*sqrt(6*n) - n) * n^(n + 1/4) / (2^(3/4)*3^(1/4)) * (1 + 97/(16*sqrt(6*n))). - _Vaclav Kotesovec_, Nov 13 2017
%F A289212 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 6^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p A289212 a:= n-> n! * add(binomial(n, i)*6^i/i!, i=0..n):
%p A289212 seq(a(n), n=0..20);
%t A289212 Table[n!*LaguerreL[n, -6], {n, 0, 20}] (* _Indranil Ghosh_, Jul 04 2017 *)
%o A289212 (Python)
%o A289212 from mpmath import *
%o A289212 mp.dps=100
%o A289212 def a(n): return int(fac(n)*laguerre(n, 0, -6))
%o A289212 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Jul 04 2017
%o A289212 (PARI) my(x = 'x + O('x^30)); Vec(serlaplace(exp(6*x/(1-x))/(1-x))) \\ _Michel Marcus_, Jul 04 2017
%o A289212 (PARI) a(n) = n!*pollaguerre(n, 0, -6); \\ _Michel Marcus_, Feb 05 2021
%Y A289212 Column k=6 of A289192.
%Y A289212 Cf. A160607, A160608.
%K A289212 nonn
%O A289212 0,2
%A A289212 _Alois P. Heinz_, Jun 28 2017