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A295382 Expansion of e.g.f. exp(-2*x/(1 - x))/(1 - x).

%I A295382 #23 Jul 08 2025 10:41:34
%S A295382 1,-1,-2,-2,8,88,592,3344,14464,2944,-1121536,-21603584,-317969408,
%T A295382 -4202380288,-51322677248,-562045749248,-4751724347392,-3419742961664,
%U A295382 1260396818661376,45221885372727296,1218206507254153216,29421299633821057024,669044215287581769728,14528992234596624498688
%N A295382 Expansion of e.g.f. exp(-2*x/(1 - x))/(1 - x).
%H A295382 G. C. Greubel, <a href="/A295382/b295382.txt">Table of n, a(n) for n = 0..449</a>
%H A295382 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295382 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295382 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295382 E.g.f.: exp(-2*x/(1 - x))/(1 - x).
%F A295382 a(n) = n!*Laguerre(n,2).
%F A295382 a(n) = n!*Sum_{k=0..n} (-1)^k*binomial(n,k)*2^k/k!.
%F A295382 a(n) = n!*A160623(n)/A160624(n).
%F A295382 a(n) = Sum_{k=0..n} (-2)^(n-k)*k!*binomial(n,k)^2. - _Ridouane Oudra_, Jul 08 2025
%p A295382 a:=series(exp(-2*x/(1-x))/(1-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 27 2019
%t A295382 nmax = 23; CoefficientList[Series[Exp[-2 x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t A295382 Table[n! LaguerreL[n, 2], {n, 0, 23}]
%t A295382 Table[n! Hypergeometric1F1[-n, 1, 2], {n, 0, 23}]
%t A295382 Table[n! Sum[(-1)^k Binomial[n, k] 2^k/k!, {k, 0, n}], {n, 0, 23}]
%o A295382 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-2*x/(1-x))/(1-x))) \\ _G. C. Greubel_, Feb 06 2018
%o A295382 (Magma) [Factorial(n)*(&+[(-1)^k*Binomial(n,k)*2^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y A295382 Column k=2 of A295381.
%Y A295382 Cf. A009940, A087912, A160623, A160624, A277423.
%K A295382 sign
%O A295382 0,3
%A A295382 _Ilya Gutkovskiy_, Nov 21 2017