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

%I #5 May 10 2022 14:26:45

%S 0,1,2,9,24,110,720,5985,39200,343224,3628800,41295870,479001600,

%T 6130959120,87104969952,1318070979225,20922789888000,354344089779680,

%U 6402373705728000,121882240625961816,2432849766865689600,51041049953430700800,1124000727777607680000

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

%t nn = 23; f[x_] := Product[(1 + x^n/n!)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Exp[-x]/(1 - x), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

%Y Cf. A000166, A006973, A137852.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, May 08 2022