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A121137 Number of labeled directed multigraphs (without loops) with n arcs and no vertex of degree 0.

%I A121137 #23 Jan 26 2025 20:29:35
%S A121137 1,2,27,572,16787,631362,28980861,1570956872,98212870233,
%T A121137 6956704585554,550626446263423,48163137319172436,4613554511554200251,
%U A121137 480324019903607680066,54004504167811544647161,6521368218660772789452944,841771274136198763040518633
%N A121137 Number of labeled directed multigraphs (without loops) with n arcs and no vertex of degree 0.
%H A121137 Nathaniel Johnston, <a href="/A121137/b121137.txt">Table of n, a(n) for n = 0..50</a>
%H A121137 Math StackExchange, Marko Riedel et al., <a href="https://math.stackexchange.com/questions/5027154/">Labeled multigraphs</a>
%H A121137 Marko Riedel, <a href="/A121137/a121137.pdf">Proof of closed form by Egorychev method.</a>
%F A121137 a(n) = Sum_{m>=0} binomial(m*(m-1)+n-1,n)/2^(m+1).
%F A121137 a(n) = Sum_{m=0..2n} Sum_{q=0..m} binomial(m,q)*(-1)^(m-q)*binomial(n+q*(q-1)-1,n). - _Marko Riedel_, Jan 26 2025
%p A121137 seq(sum(binomial(m*(m-1)+n-1,n)/2^(m+1),m=0..infinity),n=0..10);
%p A121137 # alternate program
%p A121137 A121137:= n -> add(add(binomial(m, q)*(-1)^(m-q)*binomial(n+q*(q-1)-1, n), q=0..m), m=0..2*n):
%p A121137 seq(A121137(n), n=0..20); # _Marko Riedel_, Jan 26 2025
%Y A121137 Cf. A052170 (unlabeled analog), A104209, A052171.
%K A121137 easy,nonn
%O A121137 0,2
%A A121137 _Vladeta Jovovic_, Sep 06 2006