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Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + sinh(x).
9

%I #11 May 08 2022 08:45:08

%S 1,0,1,-4,21,-126,1023,-8240,84745,-864370,10925883,-133566808,

%T 1994183205,-28455880012,489891177051,-8112780640000,158096182329585,

%U -2911196026492074,64115697136312563,-1328879415116924744,31920276313015362525,-728711636884140292372

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

%t nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

%Y Cf. A006973, A137852, A353607, A353609, A353610, A353611.

%K sign

%O 1,4

%A _Ilya Gutkovskiy_, May 07 2022