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A289213 a(n) = n! * Laguerre(n,-7).

%I A289213 #27 Feb 16 2025 08:33:48
%S A289213 1,8,79,916,12113,179152,2921911,51988748,1000578817,20686611736,
%T A289213 456805020959,10721879413252,266382974861521,6980304560060384,
%U A289213 192311632290456007,5555079068684580988,167822887344661475969,5290815252203206305832,173713426149927498289903
%N A289213 a(n) = n! * Laguerre(n,-7).
%H A289213 Alois P. Heinz, <a href="/A289213/b289213.txt">Table of n, a(n) for n = 0..432</a>
%H A289213 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289213 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289213 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289213 E.g.f.: exp(7*x/(1-x))/(1-x).
%F A289213 a(n) = n! * Sum_{i=0..n} 7^i/i! * binomial(n,i).
%F A289213 a(n) = n! * A160607(n)/A160608(n).
%F A289213 a(n) ~ exp(-7/2 + 2*sqrt(7*n) - n) * n^(n + 1/4) / (sqrt(2)*7^(1/4)) * (1 + 367/(48*sqrt(7*n))). - _Vaclav Kotesovec_, Nov 13 2017
%F A289213 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 7^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p A289213 a:= n-> n! * add(binomial(n, i)*7^i/i!, i=0..n):
%p A289213 seq(a(n), n=0..20);
%t A289213 Table[n!*LaguerreL[n, -7], {n, 0, 20}] (* _Indranil Ghosh_, Jul 04 2017 *)
%o A289213 (Python)
%o A289213 from mpmath import *
%o A289213 mp.dps=100
%o A289213 def a(n): return int(fac(n)*laguerre(n, 0, -7))
%o A289213 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Jul 04 2017
%o A289213 (PARI) x = 'x + O('x^30); Vec(serlaplace(exp(7*x/(1-x))/(1-x))) \\ _Michel Marcus_, Jul 04 2017
%o A289213 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(7*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 13 2018
%Y A289213 Column k=7 of A289192.
%Y A289213 Cf. A160607, A160608.
%K A289213 nonn
%O A289213 0,2
%A A289213 _Alois P. Heinz_, Jun 28 2017