A337824 a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).

0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600

Offset: 0

Author

Ilya Gutkovskiy, Sep 24 2020

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..257

Crossrefs

Cf. A002190, A033464, A101981, A337590, A337591, A337825.

Programs

Maple

S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):
0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # Robert Israel, Jan 07 2024

Mathematica

a[0] = 0; a[n_] := a[n] = n^2 - (1/n) * Sum[(Binomial[n, k] (n - k))^2 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Log[1 + x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

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