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A009940 a(n) = n!*L_{n}(1), L_{n}(x) the n-th Laguerre polynomial. 12

%I

%S 1,0,-1,-4,-15,-56,-185,-204,6209,112400,1520271,19165420,237686449,

%T 2944654296,36392001815,441823808804,5066513855745,49021548330016,

%U 202510138910239,-8592616658156580,-399625593156546319

%N a(n) = n!*L_{n}(1), L_{n}(x) the n-th Laguerre polynomial.

%C 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 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 G. C. Greubel, <a href="/A009940/b009940.txt">Table of n, a(n) for n = 0..449</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>.

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

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

%F a(n) = 2*(n-1)*a(n-1)-(n-1)^2*a(n-2) for n>1, a(0)=1, a(1)=0.

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

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

%e 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 seq(coeff(series(exp(x/(x-1))/(1-x),x,50),x,i)*i!, i=0..20);

%p 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 with(orthopoly): seq(n!*L(n,1), n=0..20); # C. Ronaldo, Dec 19 2004

%t (NestList[ #-Integrate[ #, x ]&, 1, 32 ]/.x:>1) Range[ 0, 32 ]!

%t Table[ n! LaguerreL[ n, 1 ], {n, 18} ]

%t Table[n! Sum[(-1)^k Binomial[n, k]/k!, {k, 0, n}], {n, 0, 10}] (* _Benedict W. J. Irwin_, Apr 20 2017 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x))) \\ _G. C. Greubel_, Feb 05 2018

%o (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 Row sums of A021009.

%K sign

%O 0,4

%A _Wouter Meeussen_

%E New name using a formula from W. Meeussen's program by _Peter Luschny_, Jan 09 2015

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Last modified February 19 12:09 EST 2019. Contains 320310 sequences. (Running on oeis4.)