A336209 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

1, -1, 1, 2, -15, -46, 880, 5837, -132783, -2109238, 35966256, 1440196097, -8909037720, -1504006716551, -16097564749643, 2021100230840147, 83130656159529937, -2475528081920694566, -331363460045748820376, -3874344448291316066455, 1255007424437046915956520

Offset: 0

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

• sign

Crossrefs

Cf. A000587, A023998, A336210.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[-Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

a(n) = (n!)^2 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^2).

a(n) = (n!)^2 * [x^n] exp(1 - BesselI(0,2*sqrt(x))).