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A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

%I A319729 #15 Dec 26 2020 23:53:50
%S A319729 1,1,1,1,3,1,1,9,7,1,1,25,37,5,1,1,75,207,85,21,1,1,231,1347,525,591,
%T A319729 7,1,1,763,10125,21385,23551,3535,113,1,1,2619,86173,180201,1216701,
%U A319729 31647,30997,9,1,1,9495,819133,12066705,77636583,66620631,11485825,286929,955,1
%N A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.
%H A319729 Andrew Howroyd, <a href="/A319729/b319729.txt">Table of n, a(n) for n = 1..300</a>
%F A319729 T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - _Andrew Howroyd_, Dec 26 2020
%e A319729 Triangle begins:
%e A319729   1
%e A319729   1       1
%e A319729   1       3       1
%e A319729   1       9       7       1
%e A319729   1      25      37       5       1
%e A319729   1      75     207      85      21       1
%e A319729   1     231    1347     525     591       7       1
%e A319729   1     763   10125   21385   23551    3535     113       1
%e A319729   1    2619   86173  180201 1216701   31647   30997       9       1
%t A319729 Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]
%Y A319729 Row sums are A322555.
%Y A319729 Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.
%K A319729 nonn,tabl
%O A319729 1,5
%A A319729 _Gus Wiseman_, Dec 17 2018