A322456 Triangle of Touchard's chord enumerating polynomial coefficients [x^k] P_n(x).

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, 1, 8, 36, 120, 330, 792, 1716, 3416, 6308, 10872, 17564, 26664, 38030, 50864, 63580, 73848, 78880, 76296, 65076, 46512, 25194, 7752

Offset: 0

Author

R. J. Mathar, Dec 09 2018

Keywords

• nonn
• tabf

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

Examples

			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;
		

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).

Crossrefs

Cf. A067311, A322398 (S_n(x)), A000699 (row sums), A232223 (transpose).

Programs

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

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

Extensions

a(0)=1 prepended by Andrew Howroyd, Nov 22 2024