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A000023
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E.g.f.: exp(-2*x)/(1-x).
(Formerly M0373 N0140)
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9
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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
(list;
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history;
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internal format)
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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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T. D. Noe, Table of n, a(n) for n=0..100
A. R. Kr\"auter, Permanenten - Ein kurzer \"Uberblick
A. R. Kr\"auter, \"Uber die Permanente gewisser zirkul\"arer Matrizen...
_Simon Plouffe_, Table for n=0..2429
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FORMULA
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a(n) = Sum(k=0..n, A008290(n, k)*(-1)^k ). - Philippe Deléham, Dec 15 2003
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)=sum_{i=0..n} A008290(i)(-1)^i. - Nour-Eddine Fahssi, Jan 27 2008
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]
G.f.: hypergeom([1,k],[],x/(1+2*x))/(1+2*x) with k=1,2,3 is the generating function for A000023, A087981, and A052124 - Mark van Hoeij, Nov 08 2011
Conjecture: -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
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MAPLE
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a:=n->n!*sum(((-2)^k/k!), k=0..n): seq(a(n), n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2007
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MATHEMATICA
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FoldList[#1*#2 + (-2)^#2 &, 1, Range[22]] (* Robert G. Wilson v, Jul 07 2012 *)
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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
-- James Spahlinger, Oct 08 2012
(Sage)
@CachedFunction
def A000023(n):
if n == 0 : return 1
return n*A000023(n-1)+(-2)^n
[A000023(i) for i in (0..22)] # Peter Luschny, Oct 17 2012
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CROSSREFS
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Cf. A087891, A008290 A089258.
Sequence in context: A037223 A066988 A100384 * A010584 A131659 A137726
Adjacent sequences: A000020 A000021 A000022 * A000024 A000025 A000026
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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