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Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tanh(x).
6

%I #6 May 22 2022 08:54:58

%S 1,0,-2,8,-24,-16,-720,12032,0,-7936,-3628800,-58190848,-479001600,

%T -22368256,87178291200,6174957043712,-20922789888000,47215125069824,

%U -6402373705728000,-164824694455533568,2432902008176640000,-4951498053124096,-1124000727777607680000

%N Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + tanh(x).

%F E.g.f.: Sum_{k>=1} A067856(k) * log(1 + tanh(x^k)) / k.

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = 2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1]/((n + 1) n!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

%Y Cf. A000182, A067856, A353779, A353912, A354066, A354171, A354172, A354173, A354174, A354175.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, May 18 2022