OFFSET
0,5
COMMENTS
T(n,k) is the number of topologically connected chord diagrams with n chords and k crossings. A chord diagram is topologically connected if the graph whose vertices are the chords and whose edges are crossing pairs of chords. Note that the rows are ordered by descending order of the number of crossings and the first entry in each row corresponds with the case of k = binomial(n,2). - Andrew Howroyd, Nov 22 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..4090 (rows 0..30)
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25, P_n(x).
FORMULA
T(n,k) = [x^k] P_n(x), k=n*(n-1)/2 down to k=n-1.
G.f.: A(x,y) satisfies: A(x*B(x,y)^2,y) = B(x,y) where B(x,y) is the g.f. of A067311. - Andrew Howroyd, Nov 22 2024
EXAMPLE
The triangle starts
1;
1;
1;
1, 3;
1, 4, 10, 12;
1, 5, 15, 35, 60, 77, 55;
1, 6, 21, 56, 126, 240, 391, 546, 624, 546, 273;
1, 7, 28, 84, 210, 462, 910, 1619, 2618, 3857, 5138, 6125, 6405, 5600, 3740, 1428;
PROG
(PARI) \\ M(n) is the n-th row of A067311 as a polynomial.
M(n)={1/(1-y)^n*sum(k=0, n, (-1)^k * ( binomial(2*n, n-k)-binomial(2*n, n-k-1)) * y^(k*(k+1)/2) )}
RowGfs(n)={my(g=sum(k=0, n, M(k)*x^k, O(x*x^n))); Vec(sqrt((x/serreverse( x*g^2 ))))}
{ my(A=RowGfs(7)); for(i=1, #A, print(Vec(A[i]/y^max(0, i-2)))) } \\ Andrew Howroyd, Nov 22 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
R. J. Mathar, Dec 09 2018
EXTENSIONS
a(0)=1 prepended by Andrew Howroyd, Nov 22 2024
STATUS
approved