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

%I A009940 #63 Jul 18 2025 10:33:33
%S A009940 1,0,-1,-4,-15,-56,-185,-204,6209,112400,1520271,19165420,237686449,
%T A009940 2944654296,36392001815,441823808804,5066513855745,49021548330016,
%U A009940 202510138910239,-8592616658156580,-399625593156546319,-12547977062999220760,-350780150063930345241,-9361105704321312342764
%N A009940 a(n) = n!*L_{n}(1), where L_{n}(x) is the n-th Laguerre polynomial.
%C A009940 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!.
%C A009940 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
%H A009940 G. C. Greubel, <a href="/A009940/b009940.txt">Table of n, a(n) for n = 0..449</a>
%H A009940 Anne-Maria Ernvall-Hytönen and Tapani Matala-aho, <a href="https://arxiv.org/abs/2310.11468">Explicit estimates for the sum Sum_{k=0..n}  k!*binomial(n,k)^2*(-1)^k</a>, arXiv:2310.11468 [math.NT], 2023.
%H A009940 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>.
%F A009940 From C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004: (Start)
%F A009940 E.g.f.: exp(x/(x-1))/(1-x).
%F A009940 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.
%F A009940 a(n) = n!*Laguerre(n, 1). (End)
%F A009940 a(n) = n!*Sum_{k=0..n} (-1)^k*Binomial(n,k)/k!. - _Benedict W. J. Irwin_, Apr 20 2017
%F A009940 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
%F A009940 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselJ(0,2*sqrt(x)). - _Ilya Gutkovskiy_, Jul 17 2020
%F A009940 From _Peter Bala_, Mar 12 2023: (Start)
%F A009940 a(n) = n! * [x^n] (1 + x)^n*exp(-x).
%F A009940 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)
%F A009940 a(n) = Sum_{k=0..n} (-1)^(n-k)*k!*binomial(n,k)^2. - _Ridouane Oudra_, Jul 03 2025
%F A009940 a(n) = (-1)^n*U(-n, 1, 1)  where U(a, b, c) is the confluent hypergeometric function. - _Stefano Spezia_, Jul 04 2025
%e A009940 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, ...
%p A009940 seq(coeff(series(exp(x/(x-1))/(1-x),x,50),x,i)*i!, i=0..20);
%p A009940 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);
%p A009940 with(orthopoly): seq(n!*L(n,1), n=0..20); # C. Ronaldo, Dec 19 2004
%t A009940 (NestList[ #-Integrate[ #, x ]&, 1, 32 ]/.x:>1) Range[ 0, 32 ]!
%t A009940 Table[ n! LaguerreL[ n, 1 ], {n, 18} ]
%t A009940 Table[n! Sum[(-1)^k Binomial[n, k]/k!, {k, 0, n}], {n, 0, 10}] (* _Benedict W. J. Irwin_, Apr 20 2017 *)
%o A009940 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x/(x-1))/(1-x))) \\ _G. C. Greubel_, Feb 05 2018
%o A009940 (PARI) a(n) = n!*pollaguerre(n, 0, 1); \\ _Michel Marcus_, Feb 06 2021
%o A009940 (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
%Y A009940 Row sums of A021009.
%K A009940 sign,easy
%O A009940 0,4
%A A009940 _Wouter Meeussen_
%E A009940 New name using a formula from W. Meeussen's program by _Peter Luschny_, Jan 09 2015