A317096 Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)).

1, 0, 1, 8, 69, 704, 8485, 118824, 1900297, 34191296, 683657001, 15038537480, 360903291661, 9383240195328, 262727926084429, 7881806223689384, 252217461390469905, 8575390623429206144, 308714050531090308817, 11731134397549023854856, 469245396934886909801941, 19708307298664103361642560

Offset: 0

Author

Ilya Gutkovskiy, Aug 01 2018

Keywords

• nonn

Comments

Lah transform of A000166.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• N. J. A. Sloane, Transforms
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000166, A002866, A052852, A059115.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 26);
Coefficients(R!(Laplace( ((1-x)/(1-2*x))*Exp(x/(x-1)) ))); // G. C. Greubel, Mar 09 2021

Maple

a:=series(exp(x/(x - 1))*(1 - x)/(1 - 2*x), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 21; CoefficientList[Series[Exp[x/(x - 1)] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Subfactorial[k] n!/k!, {k, 0, n}], {n, 0, 21}]
A317096[n_]:= A317096[n]= n!*(LaguerreL[n,1] + Sum[2^(n-j-2)*LaguerreL[j,1], {j,0, n-2}]); Table[A317096[n], {n,0,25}] (* G. C. Greubel, Mar 09 2021 *)

Sage

def A317096(n): return factorial(n)*( gen_laguerre(n,0,1) + sum(2^(n-j-2)*gen_laguerre(j,0,1) for j in (0..n-2)) )
[A317096(n) for n in (0..25)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000166(k)*n!/k!.

a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n+1). - Vaclav Kotesovec, Mar 26 2019

a(n) = n!*(LaguarreL(n,1) + Sum_{j=0..n-2} 2^(n-j-2)*LaguerreL(j,1)). - G. C. Greubel, Mar 09 2021