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%I #26 Dec 02 2023 01:39:26
%S 1,3,8,19,41,84,171,347,690,1385,2825,5438,11077,24535,33720,102623,
%T 350605,-1120228,5876775,11232063,-256532422,1748895117,-4057110163,
%U -42841409122,605093026361,-3691581277925,3538657621384,186391745956155,-2296017574506751
%N Binomial transform of A109747.
%C Equals triple binomial transform of A014182.
%H G. C. Greubel, <a href="/A153732/b153732.txt">Table of n, a(n) for n = 0..500</a>
%F A007318 * A109747.
%F E.g.f.: exp(2*x+1-exp(-x)) = 1+3*x+8*x^2/2!+19*x^3/3!+....
%F a(n) = exp(1)*Sum_{k >= 0} (-1)^k*(2-k)^n/k!. Cf. A126617. - _Peter Bala_, Oct 28 2011.
%F G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+k*x-2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 17 2013
%F a(0) = 1; a(n) = 2*a(n-1) - Sum_{k=1..n} (-1)^k * binomial(n-1,k-1) * a(n-k). - _Ilya Gutkovskiy_, Dec 01 2023
%e a(3) = 19 = (1, 3, 3, 1) dot (1, 2, 3, 3) = (1 + 6 + 9 + 3); where A109747 = (1, 2, 3, 3, 2, 3, 5, -4, 5, 55, -212, ...).
%t Join[{1}, Rest[CoefficientList[Series[Exp[2*x + 1 - Exp[-x]], {x, 0, 50}], x]*Range[0, 50]!]] (* _G. C. Greubel_, Aug 31 2016 *)
%Y Cf. A109747, A014182, A126617.
%K sign
%O 0,2
%A _Gary W. Adamson_, Dec 31 2008
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