A317364 - OEIS

%I #18 Feb 23 2021 08:31:40

%S 1,2,0,-4,16,-48,64,800,-12288,127232,-1150976,9266688,-58726400,

%T 68777984,7510646784,-207794409472,4241007640576,-77359570944000,

%U 1321952191971328,-21274345818161152,313768799799607296,-3838962981483839488,21775623343518515200,859024717017756205056

%N Expansion of e.g.f. exp(2*x/(1 + x)).

%C Inverse Lah transform of the powers of 2 (A000079).

%H Robert Israel, <a href="/A317364/b317364.txt">Table of n, a(n) for n = 0..451</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F E.g.f.: Product_{k>=1} exp(-2*(-x)^k).

%F a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).

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

%F (n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - _Robert Israel_, Aug 18 2019

%F From _G. C. Greubel_, Feb 23 2021: (Start)

%F a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).

%F a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)

%p a:= proc(n) option remember; add((-1)^(n-k)*

%p       n!/k!*binomial(n-1, k-1)*2^k, k=0..n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 26 2018

%t nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]

%t Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]

%o (Sage) [1 if n==0 else (-1)^n*factorial(n)*gen_laguerre(n, -1, 2) for n in (0..25)] # _G. C. Greubel_, Feb 23 2021

%o (Magma) [n eq 0 select 1 else (-1)^n*Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), 2): n in [0..25]]; // _G. C. Greubel_, Feb 23 2021

%o (PARI) a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ _Michel Marcus_, Feb 23 2021

%Y Cf. A000079, A052897, A111884.

%Y Cf. A001263, A008297, A086915.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Jul 26 2018