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%I #14 Dec 12 2023 18:38:13
%S 1,1,3,17,155,2079,38629,951187,29979753,1175837345,56066617331,
%T 3187704802281,212628685506643,16413606252207007,1449425836362499605,
%U 144977415195565990619,16285937949513614300369,2039447464767566886933057,282862729890000953318773603
%N E.g.f.: exp(sum(bell(n)*z^n/n, n=1..infinity)).
%C The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
%C Some transform pairs, see the formula section, are: x(n) = A000027(n) and a(n) = A000262(n); x(n) = A000045(n) and a(n) = A244430(n); x(n) = A000079(n) and a(n) = A000165(n); x(n) = A000108(n) and a(n) = A213507(n); x(n) = A000142(n) and a(n) = A158876(n); x(n) = A000203(n) and a(n) = A053529(n).
%F a(n) = n! * P(n), with P(n) = (1/n)*(sum(x(n-k) * P(k), k=0..n-1)), n >=1 and P(0) = 1, with x(n) = A000110(n), the Bell numbers.
%F E.g.f.: exp(sum(x(n)*z^n/n, n=1..infinity)) with x(n) = A000110(n).
%p a := proc(n): n!*P(n) end: P := proc(n): if n=0 then 1 else P(n):= expand((1/n)*(add(x(n-k) * P(k), k=0..n-1))) fi; end: with(combinat): x := proc(n): bell(n) end: seq(a(n), n=0..18);
%t nmax = 20; CoefficientList[Series[E^(Sum[BellB[n]*z^n/n, {n, 1, nmax}]), {z, 0, nmax}], z] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jun 29 2016 *)
%Y Cf. A036039, A000110, A000165, A000262, A053529, A158876, A213507, A244430.
%K nonn
%O 0,3
%A _Johannes W. Meijer_, Jun 29 2016
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