A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

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

Alois P. Heinz, Table of n, a(n) for n = 0..2127

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

Cf. A000041, A001372, A002861, A000081.

Sequence in context: A026153 A025180 A201967 * A111017 A116408 A015518

Adjacent sequences: A116947 A116948 A116949 * A116951 A116952 A116953

Keyword

easy,nice,nonn

Author

Franklin T. Adams-Watters, Mar 29 2006

Status

approved