A173313 Partial sums of A000273.

1, 2, 5, 21, 239, 9847, 1550791, 883584231, 1794242777079, 13029751067176631, 341273461704039756983, 32523250658517590150954423, 11366777954076059092024044958647, 14669097059490883945096188099361179575, 70315671284332059012269451652168003452397495

Offset: 0

Author

Jonathan Vos Post, Feb 16 2010

Keywords

• nonn

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

Examples

			a(12) = 1 + 1 + 3 + 16 + 218 + 9608 + 1540944 + 882033440 + 1793359192848 + 13027956824399552 + 341260431952972580352 + 32522909385055886111197440 + 11366745430825400574433894004224.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A000273.

Programs

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]

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