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A000023
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Expansion of e.g.f. exp(-2*x)/(1-x).
(Formerly M0373 N0140)
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28
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1, -1, 2, -2, 8, 8, 112, 656, 5504, 49024, 491264, 5401856, 64826368, 842734592, 11798300672, 176974477312, 2831591702528, 48137058811904, 866467058876416, 16462874118127616, 329257482363600896, 6914407129633521664, 152116956851941670912
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OFFSET
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0,3
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COMMENTS
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A010843, A000023, A000166, A000142, A000522, A010842, A053486, A053487 are successive binomial transforms with the e.g.f. exp(k*x)/(1-x) and recurrence b(n) = n*b(n-1)+k^n and are related to incomplete gamma functions at k. In this case k=-2, a(n) = n*a(n-1)+(-2)^n = Gamma(n+1,k)*exp(k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*i^(n-i)*(i+k)^i. - Vladeta Jovovic, Aug 19 2002
a(n) is the permanent of the n X n matrix with -1's on the diagonal and 1's elsewhere. - Philippe Deléham, Dec 15 2003
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-2)^(n-k)*n!/(n-k)! = Sum_{k=0..n} binomial(n, k)*k!*(-2)^(n-k). - Paul Barry, Aug 26 2004
a(n) = exp(-2)*Gamma(n+1,-2) (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009
a(n) = b such that (-1)^n*Integral_{x=0..2} x^n*exp(x) dx = c + b*exp(2). - Francesco Daddi, Aug 01 2011
D-finite with recurrence: - a(n) + (n-2)*a(n-1) + 2*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
E.g.f.: 1/E(0) where E(k) = 1 - x/(1-2/(2-(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k-1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n, k)*!k, where !k is the subfactorial A000166. a(n) = (-2)^n*hypergeom([1, -n], [], 1/2). - Vladimir Reshetnikov, Oct 18 2015
For n >= 3, a(n) = r - (-1)^n mod((-1)^n r, 2^(n - floor((2/n) + log_2(n)))) where r = {n! * e^(-2) - (-2)^(n+1)/(n+1)}. - Stan Wagon, May 02 2016
0 = +a(n)*(+4*a(n+1) -2*a(n+3)) + a(n+1)*(+4*a(n+1) +3*a(n+2) -a(n+3)) +a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, Nov 20 2018
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EXAMPLE
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G.f. = 1 - x + 2*x^2 - 2*x^3 + 8*x^4 + 8*x^5 + 112*x^6 + 656*x^7 + ... - Michael Somos, Nov 20 2018
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MAPLE
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a := n -> n!*add(((-2)^k/k!), k=0..n): seq(a(n), n=0..27); # Zerinvary Lajos, Jun 22 2007
seq(simplify(KummerU(-n, -n, -2)), n = 0..22); # Peter Luschny, May 10 2022
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MATHEMATICA
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With[{r = Round[n!/E^2 - (-2)^(n + 1)/(n + 1)]}, r - (-1)^n Mod[(-1)^n r, 2^(n + Ceiling[-(2/n) - Log[2, n]])]] (* Stan Wagon May 02 2016 *)
a[n_] := (-1)^n x D[1/x Exp[x], {x, n}] x^n Exp[-x]
Table[a[n] /. x -> 2, {n, 0, 22}](* Gerry Martens , May 05 2016 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(-2*x+x*O(x^n))/(1-x), n))
(Haskell)
a000023 n = foldl g 1 [1..n]
where g n m = n*m + (-2)^m
(Sage)
@CachedFunction
if n == 0: return 1
(PARI) x='x+O('x^66); Vec( serlaplace( exp(-2*x)/(1-x)) ) \\ Joerg Arndt, Oct 06 2013
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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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STATUS
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approved
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