OFFSET
0,3
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
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}]]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Aug 21 2018
STATUS
approved