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%I #12 Jan 01 2024 08:34:45
%S 1,1,-3,7,1,-219,2581,-22973,162177,-554039,-10506419,343049631,
%T -6846400703,113528248237,-1609627861659,17371462450651,
%U -36066494745599,-5681921495461743,243263898097515037,-7398126521141652809,193119003246643917441,-4476119490014676723659,89171014860669488040757
%N Expansion of e.g.f. exp(x/(1 + x)^2).
%F E.g.f.: Product_{k>=1} exp((-1)^(k+1)*k*x^k).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!.
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*k*k!*binomial(n-1,k-1)*a(n-k).
%F D-finite with recurrence a(n) +(3*n-4)*a(n-1) +(n-1)*(3*n-5)*a(n-2) +(n-1)*(n-2)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Aug 20 2021
%p A318215 := proc(n)
%p add((-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!,k=0..n) ;
%p end proc:
%p seq(A318215(n),n=0..42) ; # _R. J. Mathar_, Aug 20 2021
%t nmax = 22; CoefficientList[Series[Exp[x/(1 + x)^2], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^(n - k) Binomial[n + k - 1, 2 k - 1] n!/k!, {k, 0, n}], {n, 0, 22}]
%t a[n_] := a[n] = Sum[(-1)^(k + 1) k k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
%t Join[{1}, Table[(-1)^(n + 1) n n! HypergeometricPFQ[{1 - n, 1 + n}, {3/2, 2}, 1/4], {n, 22}]]
%Y Cf. A001563, A082579, A111884, A181983.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Aug 21 2018
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