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A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).

%I A346976 #11 Jan 23 2025 08:31:10
%S A346976 1,10,65,350,1666,6510,7855,-270050,-4942894,-63052990,-682650605,
%T A346976 -6309889950,-42960995804,348211510,7739540496935,202902567668150,
%U A346976 3863986259609686,61527382177040010,807717870749781475,7066953051021894250,-33781117662453993424
%N A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).
%F A346976 a(n) = Stirling2(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
%F A346976 a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan((2*arctan(1/(1 + 1/6^(1/4)))) / log(1 + 2*6^(1/4) + 2*6^(1/2)))) / (4*arctan(1/(1 + 1/6^(1/4)))^2 + log(1 + 2*6^(1/4) + 2*6^(1/2))^2)^(n/2). - _Vaclav Kotesovec_, Aug 09 2021
%F A346976 a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling2(n,4*k)/(k * 24^k). - _Seiichi Manyama_, Jan 23 2025
%t A346976 nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
%t A346976 a[n_] := a[n] = StirlingS2[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]
%Y A346976 Cf. A000453, A327505, A346954, A346974, A346975, A346977.
%Y A346976 Cf. A346946, A346954.
%K A346976 sign
%O A346976 4,2
%A A346976 _Ilya Gutkovskiy_, Aug 09 2021