A346944 Expansion of e.g.f. log( 1 + log(1 + x)^2 / 2 ).

1, -3, 8, -20, 49, -189, 1791, -21132, 228306, -2274690, 22190772, -230289696, 2756380782, -38757988710, 608149754538, -10057914084048, 171037444641816, -3000345245061048, 55157102668064592, -1077263181846230400, 22411300073192730360, -492846784406541548280

Offset: 2

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

• sign

Crossrefs

Cf. A000254, A003713, A081048, A346945, A346946, A346947.

Cf. A346974.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS1[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]

Formula

a(n) = Stirling1(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,2) * k * a(k).

a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling1(n,2*k)/(k * 2^k). - Seiichi Manyama, Jan 23 2025