A346947 Expansion of e.g.f. log( 1 + log(1 + x)^5 / 5! ).

1, -15, 175, -1960, 22449, -269451, 3423860, -46238280, 664233856, -10143487354, 164423204582, -2823783679080, 51273355515264, -982236541934430, 19809898439192946, -419752648063849626, 9325875631405818996, -216846992855331506052, 5267598064689049209252

Offset: 5

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

• sign

Crossrefs

Cf. A000482, A003713, A346944, A346945, A346946.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS1[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 23}]

Formula

a(n) = Stirling1(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,5) * k * a(k).

a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling1(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025