A317365 Expansion of e.g.f. x*exp(x/(1 + x))/(1 + x).

0, 1, 0, -3, 16, -75, 336, -1295, 1632, 55881, -1124000, 16722981, -229985040, 3089923837, -41225160144, 545880027225, -7069180940864, 86130735547665, -882387869940288, 3847692639294541, 171852333163131600, -8392137456287472699, 276055495385982856720, -8067943451470397940543

Offset

0,4

Comments

Inverse Lah transform of the nonnegative integers (A001477).

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

N. J. A. Sloane, Transforms

Index entries for sequences related to Laguerre polynomials

Formula

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

From G. C. Greubel, Mar 05 2021: (Start)

a(n) = n! * Hypergeometric1F1([-(n-1)], [1], -1).

a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1). (End)

Maple

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/(k-1)!*binomial(n-1, k-1), k=1..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

Mathematica

nmax = 23; CoefficientList[Series[x Exp[x/(1 + x)]/(1 + x) , {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] n!/(k - 1)!, {k, n}], {n, 0, 23}]
Join[{0}, Table[(-1)^(n+1) n! LaguerreL[n-1, 1], {n, 23}]]

Prog

(Sage) [0 if n==0 else (-1)^(n+1)*factorial(n)*gen_laguerre(n-1, 0, 1) for n in (0..25)] # G. C. Greubel, Mar 05 2021
(Magma) [n eq 0 select 0 else (-1)^(n+1)*Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), 1): n in [0..25]]; // G. C. Greubel, Mar 05 2021
(PARI) a(n) = (-1)^(n+1)*n!*pollaguerre(n-1, 0, 1); \\ Michel Marcus, Mar 06 2021

Crossrefs

Cf. A001477, A009940, A052852.

Sequence in context: A221829 A004303 A335625 * A207836 A372417 A005947

Adjacent sequences: A317362 A317363 A317364 * A317366 A317367 A317368

Keyword

sign

Author

Ilya Gutkovskiy, Jul 26 2018

Status

approved