OFFSET
1,2
REFERENCES
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n=1..30
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
FORMULA
a(n) = A062738(n)/2^n, since binary relations = digraphs with loops. - Ralf Stephan and Vladeta Jovovic, Mar 24 2004
E.g.f.: log(sum n>=0, 2^(n^2-n)*x^n/n!).
a(n) = A053763(n) - (1/n) * Sum_{k=1..n-1} k*C(n,k)*a(k)*A053763(n-k). - Geoffrey Critzer, Oct 24 2012
MAPLE
b:= n-> 2^(n^2-n):
a:= proc(n) option remember; local k; `if`(n=0, 1,
b(n)- add(k*binomial(n, k) *b(n-k)*a(k), k=1..n-1)/n)
end:
seq(a(n), n=1..20); # Alois P. Heinz, Oct 21 2012
MATHEMATICA
Range[0, 20]! CoefficientList[Series[D[1 + Log[Sum[2^(n^2 - n) x^n/n!, {n, 0, 20}]], x], {x, 0, 20}], x]
c[n_]:=2^(n(n-1))-Sum[k Binomial[n, k]c[k] 2^((n-k)(n-k-1)), {k, 1, n-1}]/n; c[0]=1; Table[c[i], {i, 0, 20}] (* Geoffrey Critzer, Oct 24 2012 *)
PROG
(PARI) v=Vec(log(sum(n=0, default(seriesprecision), 2^(n^2-n)*x^n/n!))); for(i=1, #v, v[i]*=(i-1)!); v \\ Charles R Greathouse IV, Feb 14 2011
(Sage)
b = lambda n: 2^(n^2-n)
@cached_function
def A003027(n):
return b(n) - sum(k*binomial(n, k)*b(n-k)*A003027(k) for k in (1..n-1)) / n
[A003027(n) for n in (1..13)] # Peter Luschny, Jan 18 2016
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Goran Kilibarda
STATUS
approved