OFFSET
0,2
COMMENTS
Partial sums of number of directed graphs (or digraphs) with n nodes. The subsequence of primes in this partial sum begins 2, 5, 239, then no more through a(20).
a(n) is the number of isolated points over all directed graphs with (n + 1) nodes. - Geoffrey Critzer, Oct 08 2012
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..60
FORMULA
a(n) = Sum_{i=0..n} A000273(i).
O.g.f.: A(x)/(1-x) where A(x) is the o.g.f. for A000273. - Geoffrey Critzer, Oct 08 2012
EXAMPLE
a(12) = 1 + 1 + 3 + 16 + 218 + 9608 + 1540944 + 882033440 + 1793359192848 + 13027956824399552 + 341260431952972580352 + 32522909385055886111197440 + 11366745430825400574433894004224.
MAPLE
b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(p[j]-1+add(
igcd(p[k], p[j]), k=1..j-1)*2, j=1..nops(p)))([l[], 1$n])),
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
end:
a:= proc(n) option remember; b(n$2, [])+`if`(n=0, 0, a(n-1)) end:
seq(a(n), n=0..16); # Alois P. Heinz, Sep 04 2019
MATHEMATICA
nn=20; d=Sum[NumberOfDirectedGraphs[n]x^n, {n, 0, nn}]; CoefficientList[Series[d/(1-x), {x, 0, nn}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Feb 16 2010
STATUS
approved