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A009940
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a(n) = n!*L_{n}(1), where L_{n}(x) is the n-th Laguerre polynomial.
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28
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1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400, 1520271, 19165420, 237686449, 2944654296, 36392001815, 441823808804, 5066513855745, 49021548330016, 202510138910239, -8592616658156580, -399625593156546319
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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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) ~ 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
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)
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EXAMPLE
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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, ...
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MAPLE
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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
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MATHEMATICA
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(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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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New name using a formula from W. Meeussen's program by Peter Luschny, Jan 09 2015
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STATUS
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approved
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