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Expanison of e.g.f. log( 1 + log(1 + x)^3 / 3! ).
4

%I #14 Jan 23 2025 08:31:23

%S 1,-6,35,-235,1834,-16352,164044,-1830630,22513326,-302700926,

%T 4419167532,-69637654996,1178377833424,-21315571470320,

%U 410529985172400,-8388475139138320,181270810764205440,-4130796696683135280,99008773205008777760,-2490134250475836315120

%N Expanison of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

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

%F 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

%t nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &

%t 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}]

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

%Y Cf. A346975.

%K sign

%O 3,2

%A _Ilya Gutkovskiy_, Aug 08 2021