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Incomplete Gamma Function at -3.
7

%I #52 May 10 2022 14:11:16

%S 1,-2,5,-12,33,-78,261,-360,3681,13446,193509,1951452,23948865,

%T 309740922,4341155877,65102989248,1041690874689,17708615729550,

%U 318755470552389,6056352778233924,121127059051462881,2543668229620367298

%N Incomplete Gamma Function at -3.

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

%F E.g.f.: exp(-3x)/(1-x). - _Michael Somos_, Mar 06 2004

%F 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

%F G.f.: hypergeom([1,1],[],x/(1+3*x))/(1+3*x). - _Mark van Hoeij_, Nov 08 2011

%F 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

%F 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

%F 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

%F a(n) ~ n! * exp(-3). - _Vaclav Kotesovec_, Oct 08 2013

%F a(n) = (-3)^(n-1)*hypergeom([1, 1-n], [], 1/3). - _Vladimir Reshetnikov_, Oct 18 2015

%F a(n) = KummerU(-n, -n, -3). - _Peter Luschny_, May 10 2022

%p a := n -> n!*add(((-3)^(k)/k!), k=0..n): seq(a(n), n=0..21); # _Zerinvary Lajos_, Jun 22 2007

%p seq(simplify(KummerU(-n, -n, -3)), n = 0..21); # _Peter Luschny_, May 10 2022

%t Table[ Gamma[ n, -3 ]*E^(-3), {n, 1, 24} ] (* corrected by _Peter Luschny_, Oct 17 2012 *)

%t a[n_] := (-1)^n x D[1/x Exp[x], {x, n}] x^n Exp[-x]

%t Table[a[n] /. x -> 3, {n, 0, 20}] (* _Gerry Martens_ , May 05 2016 *)

%o (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(-3*x+x*O(x^n))/(1-x),n)) /* _Michael Somos_, Mar 06 2004 */

%o (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 */

%o (Sage)

%o @CachedFunction

%o def A010843(n):

%o if (n) == 1 : return 1

%o return (n-1)*A010843(n-1)+(-3)^(n-1)

%o [A010843(i) for i in (1..22)] # _Peter Luschny_, Oct 17 2012

%Y See also A000023, A000166, A000142, A000522, A010842, A053486, A053487, A080954, A080955.

%Y Cf. A008279 A008279 A089258.

%K sign

%O 1,2

%A _Simon Plouffe_