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A317363 Expansion of e.g.f. 1/(2 - exp(x/(1 + x))).

%I A317363 #7 Jul 26 2018 16:29:56
%S A317363 1,1,1,1,3,1,33,-83,955,-5243,44913,-285647,1672179,3544009,
%T A317363 -352029311,9470312053,-208005703605,4326748972141,-88602638362863,
%U A317363 1819530461684473,-37722654765171965,791428823931046321,-16784285106705759519,358449656565896328061,-7653024671576463436197
%N A317363 Expansion of e.g.f. 1/(2 - exp(x/(1 + x))).
%C A317363 Inverse Lah transform of the Fubini numbers (A000670).
%H A317363 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A317363 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*A000670(k)*n!/k!.
%p A317363 b:= proc(n) option remember; `if`(n=0, 1,
%p A317363       add(b(n-j)*binomial(n, j), j=1..n))
%p A317363     end:
%p A317363 a:= proc(n) option remember; add((-1)^(n-k)*
%p A317363       n!/k!*binomial(n-1, k-1)*b(k), k=0..n)
%p A317363     end:
%p A317363 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 26 2018
%t A317363 nmax = 24; CoefficientList[Series[1/(2 - Exp[x/(1 + x)]), {x, 0, nmax}], x] Range[0, nmax]!
%t A317363 Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] HurwitzLerchPhi[1/2, -k, 0] n!/(2 k!), {k, 0, n}], {n, 0, 24}]
%Y A317363 Cf. A000670, A084358.
%K A317363 sign
%O A317363 0,5
%A A317363 _Ilya Gutkovskiy_, Jul 26 2018