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A364344 Number of endofunctions on [n] such that the number of elements that are mapped to i is a multiple or a divisor of i.

%I A364344 #15 Oct 27 2023 12:20:41
%S A364344 1,1,4,20,177,1462,21919,254802,4816788,82401465,1929926410,
%T A364344 35256890748,1152938630784,24977973856643,823036511854847,
%U A364344 24332827884557037,954801492779273665,27023410818058291822,1309814517293654535339,41375530521928893861920
%N A364344 Number of endofunctions on [n] such that the number of elements that are mapped to i is a multiple or a divisor of i.
%H A364344 Alois P. Heinz, <a href="/A364344/b364344.txt">Table of n, a(n) for n = 0..400</a>
%e A364344 a(0) = 1: ().
%e A364344 a(1) = 1: (1).
%e A364344 a(2) = 2: (11), (12), (21), (22).
%e A364344 a(3) = 20 (111), (112), (113), (121), (122), (123), (131), (132), (211), (212), (213), (221), (223), (231), (232), (311), (312), (321), (322), (333).
%p A364344 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A364344       add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)+add(
%p A364344      `if`(d>n or d=i, 0, b(n-d, i-1)*binomial(n, d)),
%p A364344           d=numtheory[divisors](i))))
%p A364344     end:
%p A364344 a:= n-> b(n$2):
%p A364344 seq(a(n), n=0..19);
%t A364344 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]* Binomial[n, i*j], {j, 0, n/i}]+Sum[If[d>n || d == i, 0, b[n - d, i - 1]* Binomial[n, d]], {d, Divisors[i]}]]];
%t A364344 a[n_] := b[n, n];
%t A364344 Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Oct 27 2023, after _Alois P. Heinz_ *)
%Y A364344 Cf. A000312, A178682, A334370, A364327, A364328.
%K A364344 nonn
%O A364344 0,3
%A A364344 _Alois P. Heinz_, Jul 19 2023