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%I #21 Jul 31 2018 03:32:43
%S 0,1,6,51,524,6405,90834,1467319,26607384,535277961,11832460190,
%T 285105945531,7437546405156,208846303056781,6280859188226154,
%U 201420656725873215,6861297209758777904,247422535745846839569,9416194788956228294454,377156775060354873848131
%N Expansion of e.g.f.: exp(-x)*x/(1-2*x)^2.
%H G. C. Greubel, <a href="/A263895/b263895.txt">Table of n, a(n) for n = 0..401</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%F a(n) = 2^n*(2*Gamma(n+2,-1/2)-Gamma(n+1,-1/2))*exp(-1/2)/4, where Gamma(a,x) is the upper incomplete Gamma function.
%F a(n) = (A000354(n+1)-A000354(n))/4.
%F For n > 0, a(n) = (-1)^(n+1)*n*hypergeom([2,1-n],[],2).
%F Recurrence: a(0) = 0, a(1) = 1, a(2) = 6, a(n) = 2*(n+1)*a(n-1) + 3*a(n-2) + 2*(2-n)*a(n-3).
%F 0 = a(n)*(+2*a(n+1) - 3*a(n+2) - 8*a(n+3) + a(n+4)) + a(n+1)*(+3*a(n+1) + 4*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2) + 2*a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0. - _Michael Somos_, Oct 29 2015
%e G.f. = x + 6*x^2 + 51*x^3 + 524*x^4 + 6405*x^5 + 90834*x^6 + 1467319*x^7 + ...
%p seq(coeff(series(factorial(n)*exp(-x)*x/(1-2*x)^2, x,n+1),x,n),n=0..25); # _Muniru A Asiru_, Jul 31 2018
%t Round@Table[2^n (2 Gamma[n+2, -1/2] - Gamma[n+1, -1/2])/(4 Sqrt[E]), {n, 0, 20}]
%t With[{nmax = 30}, CoefficientList[Series[Exp[-x]*x/(1 - 2*x)^2, {x, 0, nmax}], x]] (* _G. C. Greubel_, Jul 31 2018 *)
%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(exp(-x)*x/(1-2*x)^2))) \\ _G. C. Greubel_, Jul 31 2018
%o (GAP) a:=[0,1,6];; for n in [4..25] do a[n]:=2*n*a[n-1]+3*a[n-2]+2*(3-n)*a[n-3]; od; a; # _Muniru A Asiru_, Jul 31 2018
%Y Cf. A000354.
%K nonn
%O 0,3
%A _Vladimir Reshetnikov_, Oct 28 2015
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