A009940 a(n) = n!*L_{n}(1), where L_{n}(x) is the n-th Laguerre polynomial.

1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400, 1520271, 19165420, 237686449, 2944654296, 36392001815, 441823808804, 5066513855745, 49021548330016, 202510138910239, -8592616658156580, -399625593156546319

Offset

0,4

Comments

Previous name was: Form the iterate f[ f[ .. f[ 1 ] ] ] or f^n [ 1 ] with f[ stuff ] defined as ( stuff - Integrate[ stuff over x ] ), set x=1 and multiply by n!.

This presumably means the recurrence L(n+1,x) = L(n,x) - Integral_{t=0..x} L(n,t) dt with L(0,x) = 1, which is satisfied by the Laguerre polynomials. - Robert Israel, Jan 09 2015

Links

G. C. Greubel, Table of n, a(n) for n = 0..449

Anne-Maria Ernvall-Hytönen and Tapani Matala-aho, Explicit estimates for the sum Sum_{k=0..n}  k!*binomial(n,k)^2*(-1)^k, arXiv:2310.11468 [math.NT], 2023.

Eric Weisstein's World of Mathematics, Laguerre Polynomial.

Formula

From C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004: (Start)

E.g.f.: exp(x/(x-1))/(1-x).

D-finite with recurrence a(n) = 2*(n-1)*a(n-1)-(n-1)^2*a(n-2) for n>1, a(0)=1, a(1)=0.

a(n) = n!*Laguerre(n, 1). (End)

a(n) = n!*Sum_{k=0..n} (-1)^k*Binomial(n,k)/k!. - Benedict W. J. Irwin, Apr 20 2017

a(n) ~ sqrt(2) * n^(n + 1/4) / exp(n - 1/2) * (sin(2*sqrt(n) + Pi/4) + (17*cos(2*sqrt(n) + Pi/4)) / (48*sqrt(n))). - Vaclav Kotesovec, Feb 25 2019

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselJ(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020

From Peter Bala, Mar 12 2023: (Start)

a(n) = n! * [x^n] (1 + x)^n*exp(-x).

a(n + k) == (-1)^k*a(n) (mod k) for all n and k. It follows that the sequence a(n) taken modulo 2*k is periodic with the period dividing 2*k. See A047974. (End)

Example

The first few f[ x ] are 1, 1 - x, 1 - 2*x + x^2/2, 1 - 3*x + (3*x^2)/2 - x^3/6, giving the values 1, 0, -1/2, -2/3, ...

Maple

seq(coeff(series(exp(x/(x-1))/(1-x), x, 50), x, i)*i!, i=0..20);
A009940:=proc(n) options remember: if n<2 then RETURN([1, 0][n+1]) else RETURN(2*(n-1)*A009940(n-1)-(n-1)^2*A009940(n-2)) fi: end; seq(A009940(n), n=0..20);
with(orthopoly): seq(n!*L(n, 1), n=0..20); # C. Ronaldo, Dec 19 2004

Mathematica

(NestList[ #-Integrate[ #, x ]&, 1, 32 ]/.x:>1) Range[ 0, 32 ]!
Table[ n! LaguerreL[ n, 1 ], {n, 18} ]
Table[n! Sum[(-1)^k Binomial[n, k]/k!, {k, 0, n}], {n, 0, 10}] (* Benedict W. J. Irwin, Apr 20 2017 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x/(x-1))/(1-x))) \\ G. C. Greubel, Feb 05 2018
(PARI) a(n) = n!*pollaguerre(n, 0, 1); \\ Michel Marcus, Feb 06 2021
(Magma) I:=[1, 0]; [n le 2 select I[n] else 2*(n-1)*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 05 2018

Crossrefs

Row sums of A021009.

Sequence in context: A047018 A064813 A183932 * A081163 A082133 A060111

Adjacent sequences: A009937 A009938 A009939 * A009941 A009942 A009943

Keyword

sign,easy

Author

Wouter Meeussen

Extensions

New name using a formula from W. Meeussen's program by Peter Luschny, Jan 09 2015

Status

approved