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A295384 a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(2*n,n-k)*n^k/k!.

%I A295384 #17 Feb 16 2025 08:33:52
%S A295384 1,1,0,-15,-112,-135,9504,152425,610560,-27692847,-765107200,
%T A295384 -6289891839,213472972800,9380264146825,129378550468608,
%U A295384 -3294028613874375,-226623617585053696,-4707649131227927775,83803818828756418560,9446689798312021406353,277055229100887244800000
%N A295384 a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(2*n,n-k)*n^k/k!.
%H A295384 G. C. Greubel, <a href="/A295384/b295384.txt">Table of n, a(n) for n = 0..400</a>
%H A295384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295384 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295384 a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x)^(n+1).
%F A295384 a(n) = n!*Laguerre(n,n,n).
%F A295384 a(n) = Pochhammer(n, n)*hypergeom([1 - n], [n], n). - _Peter Luschny_, Mar 23 2023
%p A295384 a := n -> pochhammer(n, n)*hypergeom([1 - n], [n], n):
%p A295384 seq(simplify(a(n)), n = 0..20); # _Peter Luschny_, Mar 23 2023
%t A295384 Table[n! SeriesCoefficient[Exp[-n x/(1 - x)]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 20}]
%t A295384 Table[n! LaguerreL[n, n, n], {n, 0, 20}]
%t A295384 Table[(-1)^n HypergeometricU[-n, n + 1, n], {n, 0, 20}]
%t A295384 Join[{1}, Table[n! Sum[(-1)^k Binomial[2 n, n - k] n^k/k!, {k, 0, n}], {n, 1, 20}]]
%o A295384 (PARI) for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*binomial(2*n,n-k)*n^k/k!), ", ")) \\ _G. C. Greubel_, Feb 06 2018
%o A295384 (Magma) [Factorial(n)*(&+[(-1)^k*Binomial(2*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y A295384 Cf. A006902, A277373, A277423, A295385.
%K A295384 sign
%O A295384 0,4
%A A295384 _Ilya Gutkovskiy_, Nov 21 2017