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%I #25 Nov 20 2022 02:02:05
%S 1,1,3,7,27,127,747,5167,41067,368047,3669867,40284847,482671467,
%T 6267305647,87660962667,1313941673647,21010450850667,357001369769647,
%U 6423384156578667,122002101778601647,2439325392333218667
%N Row sums of triangle A136579.
%H Seiichi Manyama, <a href="/A136580/b136580.txt">Table of n, a(n) for n = 0..449</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>.
%F G.f.: 2/(1-x^2)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 29 2013
%F G.f.: Q(0)/(1-x^2), where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 22 2013
%F From _Vladimir Reshetnikov_, Oct 29 2015: (Start):
%F a(n) = (-1)^n*exp(1)*Gamma(0, 1)/2 - Re(Gamma(0, -1))*exp(-1)/2 + (n+2)!*((-1)^n*Re(Gamma(-n-2, -1))*exp(-1)-Gamma(-n-2, 1)*exp(1))/2, where Gamma(a, x) is the upper incomplete Gamma function.
%F D-finite with recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = n*a(n-1) + a(n-2) - n*a(n-3).
%F E.g.f.: 1/(1-x) + (exp(x-1)*(Ei(1)-Ei(1-x)) + exp(1-x)*(Ei(x-1)-Ei(-1)))/2, where Ei(x) is the exponential integral.
%F a(n+1)-a(n) = A153229(n+2) = (-1)^(n+1)*A058006(n+1).
%F (End)
%F 0 = a(n)*(+a(n+1) - a(n+2) - a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - 2*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0. - _Michael Somos_, Oct 29 2015
%e a(4) = 27 = sum of row 4 terms, triangle A136579: (1 + 0 + 2 + 0 + 24) = 0! + 2! + 4!.
%e a(5) = 127 = sum of row 5 terms, triangle A136579: (0 + 1 + 0 + 6 + 0 + 120) = 1! + 3! + 5!
%e G.f. = 1 + x + 3*x^2 + 7*x^3 + 27*x^4 + 127*x^5 + 747*x^6 + 5167*x^7 + 41067*x^8 + ...
%p A136580 := proc(n)
%p add( (n-2*i)!,i=0..floor(n/2) ) ;
%p end proc: # _R. J. Mathar_, Jun 04 2021
%t a[0] = 1; a[1] = 1; a[2] = 3; a[n_] := a[n] = n a[n-1] + a[n-2] - n a[n-3]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 29 2015 *)
%Y Cf. A136579, A000142, A153229, A058006.
%Y Cf. A358498, A358499, A358500.
%K nonn,easy
%O 0,3
%A _Gary W. Adamson_, Jan 09 2008
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