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
Links
- Nathaniel Johnston, Table of n, a(n) for n = 0..60
Programs
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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
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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 *)
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