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%I #16 Jan 11 2022 02:32:28
%S 1,1,1,1,1,3,3,1,6,15,16,3,1,10,45,110,140,60,10,1,15,105,435,1125,
%T 1701,1200,480,105,10,1,21,210,1295,5355,14952,26572,26670,17535,7840,
%U 2331,420,35,1,28,378,3220,19075,81228,246414,507424,666015,620900,431368
%N Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
%H Andrew Howroyd, <a href="/A117279/b117279.txt">Table of n, a(n) for n = 0..1403</a> (rows 0..25)
%F E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).
%e Triangle begins:
%e 1;
%e 1;
%e 1, 1;
%e 1, 3, 3;
%e 1, 6, 15, 16, 3;
%e 1, 10, 45, 110, 140, 60, 10;
%e ...
%t nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Sep 05 2013 *)
%o (PARI)
%o T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
%o { my(A=T(6)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 10 2022
%Y Row sums give A047864,
%Y Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
%Y The unlabeled version is A297877.
%Y Cf. A052296, A033995.
%K nonn,tabf
%O 0,6
%A _Vladeta Jovovic_, Jun 23 2007
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