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A202477 The number of ways to build all endofunctions on each block of every set partition of {1,2,...,n}.

%I A202477 #27 Jul 13 2024 15:40:05
%S A202477 1,1,5,40,437,6036,100657,1965160,43937385,1106488720,30982333661,
%T A202477 954607270464,32090625710365,1168646120904640,45826588690845705,
%U A202477 1924996299465966976,86231288506425806033,4103067277186778016000,206655307175847710248885
%N A202477 The number of ways to build all endofunctions on each block of every set partition of {1,2,...,n}.
%H A202477 Alois P. Heinz, <a href="/A202477/b202477.txt">Table of n, a(n) for n = 0..385</a>
%F A202477 E.g.f.: exp(T(x)/(1-T(x))) where T(x) is the e.g.f. for A000169.
%F A202477 a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3) - 2/3) / sqrt(3). - _Vaclav Kotesovec_, Sep 24 2013
%F A202477 a(n) = Sum_{k=0..n} n^(n-k)*binomial(n-1,k-1)*A000262(k). - _Fabian Pereyra_, Jul 12 2024
%F A202477 The above formula can be written with the Abel polynomials: a(n) = Sum_{k=0..n} (-1)^(n - k) * A137452(n, k) * A000262(k). - _Peter Luschny_, Jul 13 2024
%p A202477 with(combinat):
%p A202477 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A202477       `if`(i<1, 0, add(i^(i*j)*b(n-i*j, i-1)*
%p A202477        multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A202477     end:
%p A202477 a:= n-> b(n$2):
%p A202477 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 29 2016
%t A202477 nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}] ;
%t A202477 Range[0, nn]! CoefficientList[Series[Exp[t/(1 - t)], {x, 0, nn}], x]
%Y A202477 Cf. A000262 (the same for permutations), A137452.
%K A202477 nonn
%O A202477 0,3
%A A202477 _Geoffrey Critzer_, Dec 19 2011