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A116950
Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.
5
1, 2, 7, 20, 61, 174, 514, 1478, 4303, 12437, 36084, 104494, 303167, 879283, 2552803, 7413583, 21544347, 62635823, 182199853, 530228946, 1543761513, 4496523995, 13102414665, 38193626823, 111375529695, 324891970936, 948051861938, 2767336312386, 8080206646244
OFFSET
0,2
COMMENTS
A001372 counts functional patterns from a set with n elements to itself; A000041 (partition function) counts functional patterns from a set with n elements to a disjoint set; this is the general case where the range may overlap the domain but may also include other values.
LINKS
FORMULA
Euler transform of A002861(n) + A000081(n+1).
a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.95576528565199497471481752412..., c = 3.435908969217935496995961718... . - Vaclav Kotesovec, Sep 10 2014
EXAMPLE
For n=2 there are the following 7 digraphs:
o-+.o-+ o->o-+ o->o o-+.o->o o->o->o o->o o->o
^.|.^.| ...^.| ^..| ^.|..... ....... ...^ ....
+-+.+-+ ...+-+ +--+ +-+..... ....... o--+ o->o
MATHEMATICA
nmax = 750;
A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt", "Table"], {_, _}][[;; nmax + 2, 2]];
A000081 = Cases[Import["https://oeis.org/A000081/b000081.txt", "Table"], {_, _}][[;; nmax + 2, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
b[n_] := A002861[[n]] + A000081[[n + 2]];
a = etr[b];
a[0] = 1;
a /@ Range[0, nmax](* Jean-François Alcover, Mar 13 2020 *)
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
STATUS
approved