OFFSET
0,2
COMMENTS
Inverse Lah transform of the powers of 2 (A000079).
LINKS
Robert Israel, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms
FORMULA
E.g.f.: Product_{k>=1} exp(-2*(-x)^k).
a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*2^k*n!/k!.
(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.
a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).
a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)
MAPLE
a:= proc(n) option remember; add((-1)^(n-k)*
n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jul 26 2018
MATHEMATICA
nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]
PROG
(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
(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
(PARI) a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jul 26 2018
STATUS
approved