A318215 Expansion of e.g.f. exp(x/(1 + x)^2).

1, 1, -3, 7, 1, -219, 2581, -22973, 162177, -554039, -10506419, 343049631, -6846400703, 113528248237, -1609627861659, 17371462450651, -36066494745599, -5681921495461743, 243263898097515037, -7398126521141652809, 193119003246643917441, -4476119490014676723659, 89171014860669488040757

Offset: 0

Author

Ilya Gutkovskiy, Aug 21 2018

Keywords

• sign

Crossrefs

Cf. A001563, A082579, A111884, A181983.

Programs

Maple

A318215 := proc(n)
    add((-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!,k=0..n) ;
end proc:
seq(A318215(n),n=0..42) ; # R. J. Mathar, Aug 20 2021

Mathematica

nmax = 22; CoefficientList[Series[Exp[x/(1 + x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n + k - 1, 2 k - 1] n!/k!, {k, 0, n}], {n, 0, 22}]
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}]
Join[{1}, Table[(-1)^(n + 1) n n! HypergeometricPFQ[{1 - n, 1 + n}, {3/2, 2}, 1/4], {n, 22}]]

Formula

E.g.f.: Product_{k>=1} exp((-1)^(k+1)*k*x^k).

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1,2*k-1)*n!/k!.

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*k*k!*binomial(n-1,k-1)*a(n-k).

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