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%I #13 Sep 08 2022 08:46:22
%S 1,1,-3,13,-71,441,-2699,9157,206193,-8443151,236126701,-6169406979,
%T 161388751657,-4327824442967,120012465557349,-3450029411174219,
%U 102741264191105761,-3160671409312412703,99982488984008583133,-3230094912866216253971,105481073534842477321881,-3423260541695907002392679
%N Expansion of e.g.f. exp(x/(1 + 2*x)).
%H G. C. Greubel, <a href="/A318223/b318223.txt">Table of n, a(n) for n = 0..250</a>
%F E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).
%F a(n) = Sum_{k=0..n} (-2)^(n-k)*binomial(n-1,k-1)*n!/k!.
%F a(0) = 1; a(n) = Sum_{k=1..n} (-2)^(k-1)*k!*binomial(n-1,k-1)*a(n-k).
%p seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # _Paolo P. Lava_, Jan 09 2019
%t nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
%t a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
%t Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ _G. C. Greubel_, Feb 07 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Feb 07 2019
%o (Sage) m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Feb 07 2019
%Y Cf. A002866, A025168, A111884, A122803.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Aug 21 2018