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A095693 Triangle read by rows: T(n,k) is the number of multigraphs without loops on n labeled nodes with k edges and maximum degree 2.

%I A095693 #7 Nov 07 2019 19:25:17
%S A095693 1,1,0,1,1,1,1,3,6,1,1,6,21,22,6,1,10,55,130,130,22,1,15,120,485,1005,
%T A095693 822,130,1,21,231,1400,4830,8547,6202,822,1,28,406,3416,17465,52052,
%U A095693 81676,52552,6202,1,36,666,7392,52101,230832,610932,859932,499194,52552
%N A095693 Triangle read by rows: T(n,k) is the number of multigraphs without loops on n labeled nodes with k edges and maximum degree 2.
%C A095693 Sum of the each row of the triangle corresponds to sequence A000985. The diagonal of the triangular array T(n,1) represents the triangular numbers (A000217). The T(n,2) diagonal represents the doubly triangular numbers (A002817).
%C A095693 Number of symmetric n X n matrices with nonnegative integer entries and all row sums 2 and trace 2*(n-k). - _Andrew Howroyd_, Nov 07 2019
%D A095693 Horne, Nicholas S. "Analysis of Viable Network Configurations from a Combinatorial, Graphical and Algebraic Perspective." Diss. Providence College, 2004.
%H A095693 Andrew Howroyd, <a href="/A095693/b095693.txt">Table of n, a(n) for n = 0..1325</a>
%F A095693 E.g.f.: sqrt(1/(1-x*y)) * exp(x + (x^2*y/(1-x*y) - x*y)/2 + x^2*y^2/4). - _Andrew Howroyd_, Nov 07 2019
%e A095693 Triangle begins:
%e A095693   1;
%e A095693   1,  0;
%e A095693   1,  1,   1;
%e A095693   1,  3,   6,    1;
%e A095693   1,  6,  21,   22,    6;
%e A095693   1, 10,  55,  130,  130,   22;
%e A095693   1, 15, 120,  485, 1005,  822,  130;
%e A095693   1, 21, 231, 1400, 4830, 8547, 6202, 822;
%e A095693   ...
%e A095693 T(3,2)=6 since there are six ways that a multigraph with 3 nodes can be constructed with 2 edges such that no vertex has degree greater than two.
%o A095693 (PARI)
%o A095693 T(n)={my(v=Vec(serlaplace(sqrt(1/(1-x*y) + O(x*x^n))*exp(x + (x^2*y/(1-x*y) - x*y)/2 + x^2*y^2/4 + O(x*x^n))))); vector(#v, i, Vecrev(v[i], i))}
%o A095693 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Nov 07 2019
%Y A095693 Row sums are A000985.
%Y A095693 Main diagonal is A002137.
%Y A095693 Columns include A000217, A002817.
%K A095693 nonn,tabl
%O A095693 0,8
%A A095693 Nicholas S. Horne (nickhorne(AT)cox.net), Jul 06 2004
%E A095693 Definition clarified and terms a(37) and beyond from _Andrew Howroyd_, Nov 07 2019