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Product_{n>=1} 1 / (1 - a(n)*x^n) = Sum_{n>=0} Bell(n)*x^n, where Bell = A000110.
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%I #8 May 13 2022 07:57:06

%S 1,1,3,9,34,132,610,2929,15604,87310,526274,3325946,22270254,

%T 155986944,1146627256,8787134873,70227355786,583161239732,

%U 5027823752930,44899767806134,414877525216196,3959806750825202,38996757506464858,395743830189684984,4134132167169618654,44409120984298440176

%N Product_{n>=1} 1 / (1 - a(n)*x^n) = Sum_{n>=0} Bell(n)*x^n, where Bell = A000110.

%F Conjecture: a(n) ~ Bell(n). - _Vaclav Kotesovec_, May 12 2022

%t A[m_, n_] := A[m, n] = Which[m == 1, BellB[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 26]

%Y Cf. A000110, A006177, A085686, A305846, A353831.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, May 12 2022