A346945 Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

1, -6, 35, -235, 1834, -16352, 164044, -1830630, 22513326, -302700926, 4419167532, -69637654996, 1178377833424, -21315571470320, 410529985172400, -8388475139138320, 181270810764205440, -4130796696683135280, 99008773205008777760, -2490134250475836315120

Offset: 3

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

• sign

Crossrefs

Cf. A000399, A003713, A081051, A346944, A346946, A346947.

Cf. A346975.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS1[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 22}]

Formula

a(n) = Stirling1(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,3) * k * a(k).

a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling1(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025