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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.
4
1, 1, 4, 20, 177, 1462, 21919, 254802, 4816788, 82401465, 1929926410, 35256890748, 1152938630784, 24977973856643, 823036511854847, 24332827884557037, 954801492779273665, 27023410818058291822, 1309814517293654535339, 41375530521928893861920
OFFSET
0,3
LINKS
EXAMPLE
a(0) = 1: ().
a(1) = 1: (1).
a(2) = 2: (11), (12), (21), (22).
a(3) = 20 (111), (112), (113), (121), (122), (123), (131), (132), (211), (212), (213), (221), (223), (231), (232), (311), (312), (321), (322), (333).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)+add(
`if`(d>n or d=i, 0, b(n-d, i-1)*binomial(n, d)),
d=numtheory[divisors](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..19);
MATHEMATICA
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]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 19 2023
STATUS
approved