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Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).
3

%I #7 May 23 2022 09:12:52

%S 1,0,-1,4,-11,-4,-547,7680,-7751,81744,-3258663,-9474816,-390445563,

%T 233029824,-964154427,4193551958016,-18431412645519,71090090006784,

%U -6436900596281679,17349989459410944,834261829219880829,-241960391975347200,-1149793471388581053219

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

%F E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arcsinh(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] = {1, 0, -1, 0}[[Mod[n, 4, 1]]] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

%Y Cf. A001818, A067856, A353819, A353914, A353972, A354116, A354172, A354275, A354276.

%K sign

%O 1,4

%A _Ilya Gutkovskiy_, May 22 2022