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A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

%I A293145 #23 Feb 24 2021 08:20:07
%S A293145 1,1,8,99,1696,37225,997056,31535371,1150303232,47538819729,
%T A293145 2195314048000,112032721984051,6261138045038592,380309520560089081,
%U A293145 24946892219825709056,1757549042234670166875,132356128415391650676736,10610067001068927596601889,902057202129607760380428288
%N A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).
%H A293145 Seiichi Manyama, <a href="/A293145/b293145.txt">Table of n, a(n) for n = 0..356</a>
%H A293145 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A293145 a(n) = n! * [x^n] Product_{k>=1} exp(n*x^k).
%F A293145 a(n) ~ exp(n/phi - n) * phi^(2*n) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 01 2017
%F A293145 a(n) = n! * Sum_{k=1..n} n^k * binomial(n-1,k-1)/k! for n > 0. - _Seiichi Manyama_, Feb 03 2021
%F A293145 a(n) = n! * LaguerreL(n-1, 1, -n) with a(0) = 1. - _G. C. Greubel_, Feb 23 2021
%t A293145 Table[n! SeriesCoefficient[Exp[n x/(1 - x)], {x, 0, n}], {n, 0, 18}]
%t A293145 Table[n! SeriesCoefficient[Product[Exp[n x^k], {k, 1, n}], {x, 0, n}], {n, 0, 18}]
%t A293145 Join[{1}, Table[Sum[n^k n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 18}]]
%t A293145 Join[{1}, Table[n n! Hypergeometric1F1[1 - n, 2, -n], {n, 1, 18}]]
%t A293145 Table[If[n==0, 1, n!*LaguerreL[n-1, 1, -n]], {n, 0, 20}] (* _G. C. Greubel_, Feb 23 2021 *)
%o A293145 (PARI) {a(n) = if(n==0, 1, n!*sum(k=1, n, n^k*binomial(n-1, k-1)/k!))} \\ _Seiichi Manyama_, Feb 03 2021
%o A293145 (PARI) a(n) = if (n, n! * pollaguerre(n-1, 1, -n), 1); \\ _Michel Marcus_, Feb 23 2021
%o A293145 (Sage) [1 if n==0 else factorial(n)*gen_laguerre(n-1, 1, -n) for n in (0..20)] # _G. C. Greubel_, Feb 23 2021
%o A293145 (Magma) [n eq 0 select 1 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 1), -n): n in [0..20]]; // _G. C. Greubel_, Feb 23 2021
%Y A293145 Main diagonal of A253286.
%Y A293145 Cf. A000262, A052857, A052897, A255806, A277372.
%K A293145 nonn
%O A293145 0,3
%A A293145 _Ilya Gutkovskiy_, Oct 01 2017