A000023 Expansion of e.g.f. exp(-2*x)/(1-x).
1, -1, 2, -2, 8, 8, 112, 656, 5504, 49024, 491264, 5401856, 64826368, 842734592, 11798300672, 176974477312, 2831591702528, 48137058811904, 866467058876416, 16462874118127616, 329257482363600896, 6914407129633521664, 152116956851941670912
Offset: 0
Examples
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
References
- 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).
Links
- Simon Plouffe, Table of n, a(n) for n = 0..250
- A. R. Kräuter, Permanenten - Ein kurzer Überblick, Séminaire Lotharingien de Combinatoire, B09b (1983), 34 pp.
- A. R. Kräuter, Über die Permanente gewisser zirkulanter Matrizen und damit zusammenhängender Toeplitz-Matrizen, Séminaire Lotharingien de Combinatoire, B11b (1984), 11 pp.
Crossrefs
Programs
-
Haskell
a000023 n = foldl g 1 [1..n] where g n m = n*m + (-2)^m -- James Spahlinger, Oct 08 2012
-
Maple
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
-
Mathematica
FoldList[#1*#2 + (-2)^#2 &, 1, Range[22]] (* Robert G. Wilson v, Jul 07 2012 *) 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 *)
-
PARI
a(n)=if(n<0,0,n!*polcoeff(exp(-2*x+x*O(x^n))/(1-x),n))
-
PARI
my(x='x+O('x^66)); Vec( serlaplace( exp(-2*x)/(1-x)) ) \\ Joerg Arndt, Oct 06 2013 -
Python
from sympy import exp, uppergamma def A000023(n): return exp(-2) * uppergamma(n + 1, -2) # David Radcliffe, Aug 20 2025
-
Sage
@CachedFunction def A000023(n): if n == 0: return 1 return n * A000023(n-1) + (-2)**n [A000023(i) for i in range(23)] # Peter Luschny, Oct 17 2012
Formula
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) = 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
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
a(n) = KummerU(-n, -n, -2). - Peter Luschny, May 10 2022
Comments