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.

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, 822, 130, 1, 21, 231, 1400, 4830, 8547, 6202, 822, 1, 28, 406, 3416, 17465, 52052, 81676, 52552, 6202, 1, 36, 666, 7392, 52101, 230832, 610932, 859932, 499194, 52552

Offset: 0

Author

Nicholas S. Horne (nickhorne(AT)cox.net), Jul 06 2004

Keywords

• nonn
• tabl

Comments

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).

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

Examples

			Triangle begins:
  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,  822,  130;
  1, 21, 231, 1400, 4830, 8547, 6202, 822;
  ...
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.
		

References

• Horne, Nicholas S. "Analysis of Viable Network Configurations from a Combinatorial, Graphical and Algebraic Perspective." Diss. Providence College, 2004.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

Crossrefs

Row sums are A000985.

Main diagonal is A002137.

Columns include A000217, A002817.

Programs

PARI

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))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 07 2019

Formula

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

Extensions

Definition clarified and terms a(37) and beyond from Andrew Howroyd, Nov 07 2019