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A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/k!.

%I A295385 #16 Feb 16 2025 08:33:52
%S A295385 1,3,32,579,14736,483115,19376928,918980139,50306339072,3121729082739,
%T A295385 216541483852800,16603614676249843,1394473165806440448,
%U A295385 127308860552307549531,12553171419275174137856,1329537514269062031406875,150531055969843353812533248,18143286205523964035258551651
%N A295385 a(n) = n!*Sum_{k=0..n} binomial(2*n,n-k)*n^k/k!.
%H A295385 G. C. Greubel, <a href="/A295385/b295385.txt">Table of n, a(n) for n = 0..335</a>
%H A295385 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295385 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295385 a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n+1).
%F A295385 a(n) = n!*Laguerre(n,n,-n).
%F A295385 a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^n / exp((2 - sqrt(2))*n). - _Vaclav Kotesovec_, Nov 21 2017
%t A295385 Table[n! SeriesCoefficient[Exp[n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 17}]
%t A295385 Table[n! LaguerreL[n, n, -n], {n, 0, 17}]
%t A295385 Table[(-1)^n HypergeometricU[-n, n + 1, -n], {n, 0, 17}]
%t A295385 Join[{1}, Table[n! Sum[Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 17}]]
%o A295385 (PARI) for(n=0,30, print1(n!*sum(k=0,n, binomial(2*n,n-k)*n^k/k!), ", ")) \\ _G. C. Greubel_, Feb 06 2018
%o A295385 (Magma) [Factorial(n)*(&+[Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y A295385 Cf. A082545, A277373, A277423, A295384, A295406, A295407, A295408, A295409.
%K A295385 nonn
%O A295385 0,2
%A A295385 _Ilya Gutkovskiy_, Nov 21 2017