A121933 Number of labeled digraphs with n arcs for which every vertex has indegree at least one and outdegree at least one.

1, 0, 1, 2, 18, 158, 1788, 23930, 370886, 6527064, 128542420, 2800362536, 66858556196, 1735834171276, 48689118113374, 1467253017578672, 47275138863637080, 1621757692715997136, 59013695834307968254, 2270400832166224741596, 92078072790064946096284

Offset

0,4

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..60

Formula

G.f.: Sum(Sum((-1)^(n-k)*binomial(n,k)*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(n-k),k=0..n),n=0..infinity).

a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.0722246614111436... . - Vaclav Kotesovec, May 07 2014

In closed form, c = 1/(sqrt(Pi*(1-log(2))) * log(2) * 2^(4+log(2)/2)). - Vaclav Kotesovec, May 04 2015

Maple

n:=20: t:=taylor(sum(sum((-1)^(m-k)*binomial(m, k)*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(m-k), k=0..m), m=0..n), x, n+1): seq(coeff(t, x, m), m=0..n); # Nathaniel Johnston, Apr 28 2011

Mathematica

Flatten[{1, Rest[CoefficientList[Series[Sum[Sum[(-1)^(n-k)*Binomial[n, k]*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(n-k), {k, 0, n}], {n, 1, 20}], {x, 0, 20}], x]]}] (* Vaclav Kotesovec, May 07 2014 *)

Crossrefs

Cf. A121252, A086193 (by # of nodes), A367500 (unlabeled version).

Sequence in context: A366953 A052838 A052876 * A108550 A322282 A270369

Adjacent sequences: A121930 A121931 A121932 * A121934 A121935 A121936

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 02 2006

Status

approved