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A317096 Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)). 1
1, 0, 1, 8, 69, 704, 8485, 118824, 1900297, 34191296, 683657001, 15038537480, 360903291661, 9383240195328, 262727926084429, 7881806223689384, 252217461390469905, 8575390623429206144, 308714050531090308817, 11731134397549023854856, 469245396934886909801941, 19708307298664103361642560 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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

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 *)

PROG

(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

(Magma)

R<x>:=PowerSeriesRing(Rationals(), 26);

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

CROSSREFS

Cf. A000166, A002866, A052852, A059115.

Sequence in context: A228421 A152109 A222064 * A327606 A336951 A335114

Adjacent sequences:  A317093 A317094 A317095 * A317097 A317098 A317099

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 01 2018

STATUS

approved

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)