OFFSET
0,8
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
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
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
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Nicholas S. Horne (nickhorne(AT)cox.net), Jul 06 2004
EXTENSIONS
Definition clarified and terms a(37) and beyond from Andrew Howroyd, Nov 07 2019
STATUS
approved