A364327 -id:A364327 - cp's OEIS frontend

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.

1, 1, 4, 20, 177, 1462, 21919, 254802, 4816788, 82401465, 1929926410, 35256890748, 1152938630784, 24977973856643, 823036511854847, 24332827884557037, 954801492779273665, 27023410818058291822, 1309814517293654535339, 41375530521928893861920

Offset: 0

Alois P. Heinz, Jul 19 2023

nonn

			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).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000312, A178682, A334370, A364327, A364328.

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);

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 *)

A364328 Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a prime divisor of i.

Original entry on oeis.org

1, 0, 1, 1, 6, 21, 110, 904, 4312, 74400, 731412, 5600761, 128196024, 792051157, 18696610816, 264267572121, 7136433698464, 57948743342529, 2228312959187256, 22463157401776612, 681974906329502904, 15395459281239915282, 463374873030990445252, 6091833036158810701465

Offset: 0

Alois P. Heinz, Jul 18 2023

nonn

			a(0) = 1: ().
a(2) = 1: (22).
a(3) = 1: (333).
a(4) = 6: (4422), (4242), (4224), (2442), (2424), (2244).
a(5) = 21: (55555), (44333), (43433), (43343), (43334), (34433), (34343), (34334), (33443), (33434), (33344), (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333).

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000040, A178682, A334370, A364327, A364344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
     `if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[factorset](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);

cp's OEIS Frontend

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.

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