0,3

Number of endofunctions on n points whose functional digraphs (with loops allowed) are nontrivially the directed sum of two or more digraphs of endofunctions. A002861 Number of connected functions (or mapping patterns) on n unlabeled points, or number of rings and branches with n edges. A001372 Number of mappings (or mapping patterns) from n points to themselves; number of endofunctions. a(n) is prime for n = 2, 6, 9, 18.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.

R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.399 and 41.401.

N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J, Algeb. Discrete Meth., 3 (1982), 359-368.

Eric Weisstein's World of Mathematics, Functional Graph.

a(n) = A001372(n) - A002861(n). a(n) = (Euler transform of A002861) - (CIK transform of A000081).

a(0) = 0, as the null digraph is formally neither connected nor disconnected.

a(1) = 0, as the unique endofunction on one point is the identity function on one value, and is connected.

a(2) = 1, as there are 3 endofunctions on two points, two of which are "prime endofunctions", and one of which is the direct sum of two copies of the unique endofunction on one point, namely two points-with-loops, or the identity function on two values; 3 - 2 = 1.

a(3) = A001372(3) - A002861(3) = 7 - 4 = 3.

a(4) = A001372(4) - A002861(4) = 19 - 9 = 10.

a(5) = A001372(5) - A002861(5) = 47 - 20 = 27.

a(6) = 130 - 51 = 79.

a(7) = 343 - 125 = 218.

a(8) = 951 - 329 = 622.

a(9) = 2615 - 862 = 1753.

a(10) = 7318 - 2311 = 5007.

Cf. A000081, A000273, A001372, A002861, A003027, A003085, A062738, A116950, A126285, A127909-A127915.

Sequence in context: A000471 A000501 A027251 this_sequence A005956 A059193 A037167

Adjacent sequences: A127909 A127910 A127911 this_sequence A127913 A127914 A127915

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 06 2007