A346975 Expansion of e.g.f. log( 1 + (exp(x) - 1)^3 / 3! ).

1, 6, 25, 80, 91, -1694, -22875, -193740, -1119569, -768394, 101162425, 1930987240, 23583202371, 181575384906, -306743537075, -45405986594980, -1070132302146089, -16439720013909794, -145808623945689375, 1048196472097011600, 84226169502099763051

Offset: 3

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

• sign

Crossrefs

Cf. A000392, A327504, A346974, A346976, A346977.

Cf. A346390, A346945.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

Formula

a(n) = Stirling2(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).

a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan(2*arctan(3^(5/6)/(2^(2/3) + 3^(1/3))) / log(1 + 6^(1/3) + 6^(2/3)))) / (4*arctan(3^(5/6)/(2^(2/3) + 3^(1/3)))^2 + log(1 + 6^(1/3) + 6^(2/3))^2)^(n/2). - Vaclav Kotesovec, Aug 09 2021

a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025