A010843 Incomplete Gamma Function at -3.
1, -2, 5, -12, 33, -78, 261, -360, 3681, 13446, 193509, 1951452, 23948865, 309740922, 4341155877, 65102989248, 1041690874689, 17708615729550, 318755470552389, 6056352778233924, 121127059051462881, 2543668229620367298
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Keywords
References
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Links
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Crossrefs
Programs
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Maple
a := n -> n!*add(((-3)^(k)/k!), k=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jun 22 2007 seq(simplify(KummerU(-n, -n, -3)), n = 0..21); # Peter Luschny, May 10 2022
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Mathematica
Table[ Gamma[ n, -3 ]*E^(-3), {n, 1, 24} ] (* corrected by Peter Luschny, Oct 17 2012 *) a[n_] := (-1)^n x D[1/x Exp[x], {x, n}] x^n Exp[-x] Table[a[n] /. x -> 3, {n, 0, 20}] (* Gerry Martens , May 05 2016 *) -
PARI
a(n)=if(n<0,0,n!*polcoeff(exp(-3*x+x*O(x^n))/(1-x),n)) /* Michael Somos, Mar 06 2004 */
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PARI
a(n)=local(A,p);if(n<1,n==0,A=matrix(n,n,i,j,1-3*(i==j));sum(i=1,n!,if(p=numtoperm(n,i),prod(j=1,n,A[j,p[j]])))) /* Michael Somos, Mar 06 2004 */
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Sage
@CachedFunction def A010843(n): if (n) == 1 : return 1 return (n-1)*A010843(n-1)+(-3)^(n-1) [A010843(i) for i in (1..22)] # Peter Luschny, Oct 17 2012
Formula
E.g.f.: exp(-3x)/(1-x). - Michael Somos, Mar 06 2004
a(0) = 1 and for n>0, a(n) is the permanent of the n X n matrix with -2's on the diagonal and 1's elsewhere. a(n) = Sum(k=0..n, A008290(n, k)*(-2)^k ). a(n) = Sum(k=0..n, A008279(n, k)*(-3)^(n-k) ). - Philippe Deléham, Dec 15 2003
G.f.: hypergeom([1,1],[],x/(1+3*x))/(1+3*x). - Mark van Hoeij, Nov 08 2011
E.g.f.: 1/E(0) where E(k)=1-x/(1-3/(3-(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 13 2012
G.f.: 1/Q(0), where Q(k)= 1 + 3*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k-2) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) ~ n! * exp(-3). - Vaclav Kotesovec, Oct 08 2013
a(n) = (-3)^(n-1)*hypergeom([1, 1-n], [], 1/3). - Vladimir Reshetnikov, Oct 18 2015
a(n) = KummerU(-n, -n, -3). - Peter Luschny, May 10 2022