A337590 a(0) = 0; a(n) = n - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k) * k * a(k).

0, 1, 0, -3, 28, -215, -174, 90223, -3840472, 103719537, 429704110, -357346077869, 35100093531900, -2005608652057595, -24108041118593418, 27881407632242902515, -4876442148527153942384, 474102062424164433715937, 12637408141631813073125094, -18867461801192524662360616421

Offset: 0

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

• sign

Crossrefs

Cf. A002190, A009306, A336227.

Programs

Mathematica

a[0] = 0; a[n_] := a[n] = n - (1/n) Sum[Binomial[n, k]^2 (n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Log[1 + Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + sqrt(x) * BesselI(1,2*sqrt(x))).

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n * x^n / (n!)^2).