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A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k). 3
1, 0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..444

Index entries for sequences related to Laguerre polynomials

FORMULA

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*a(n-k)/(n-k)! for n > 0.

E.g.f.: (1 - x) * exp(x/(1 - x)). - Ilya Gutkovskiy, Jul 27 2020

a(n) = (n!/(n-1))*( 2*LaguerreL(n-1, -1) - LaguerreL(n, -1) ) with a(0) = 1, a(1) = 0. - G. C. Greubel, Mar 10 2021

a(n) ~ n^(n - 3/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 7/(6*sqrt(n))). - Vaclav Kotesovec, Mar 10 2021

MATHEMATICA

Table[If[n<2, 1-n, (n!/(n-1))*(2*LaguerreL[n-1, -1] - LaguerreL[n, -1])], {n, 0, 30}] (* G. C. Greubel, Mar 10 2021 *)

PROG

(PARI) {a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*x^k/k)+x*O(x^n)), n)}

(Magma)

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;

[1, 0]cat[(Factorial(n)/(n-1))*(2*l(n-1, 0, -1) - l(n, 0, -1)): n in [2..30]]; // G. C. Greubel, Mar 10 2021

(Sage) [1-n if n<2 else (factorial(n)/(n-1))*(2*gen_laguerre(n-1, 0, -1) - gen_laguerre(n, 0, -1)) for n in (0..30)] # G. C. Greubel, Mar 10 2021

CROSSREFS

Cf. A009940, A052852, A052887, A288269.

Sequence in context: A209881 A052852 A288869 * A265952 A121124 A180399

Adjacent sequences:  A288265 A288266 A288267 * A288269 A288270 A288271

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 20 2017

STATUS

approved

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Last modified September 19 03:31 EDT 2021. Contains 347550 sequences. (Running on oeis4.)