A232223 Triangle read by rows: T(n,k) = number of connected matchings with n crossings and k chords, in a disk, k=2..n+1.

1, 0, 3, 0, 1, 12, 0, 0, 10, 55, 0, 0, 4, 77, 273, 0, 0, 1, 60, 546, 1428, 0, 0, 0, 35, 624, 3740, 7752, 0, 0, 0, 15, 546, 5600, 25194, 43263, 0, 0, 0, 5, 391, 6405, 46512, 168245, 246675, 0, 0, 0, 1, 240, 6125, 65076, 368676, 1118260, 1430715, 0, 0, 0, 0, 126, 5138, 76296, 606879, 2833600, 7413705, 8414640, 0, 0, 0, 0, 56, 3857, 78880, 834195, 5348420, 21312720, 49085400, 50067108

Offset: 1

Author

N. J. A. Sloane, Nov 22 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1,
0,3,
0,1,12,
0,0,10,55,
0,0,4,77,273,
0,0,1,60,546,1428,
0,0,0,35,624,3740,7752,
0,0,0,15,546,5600,25194,43263,
0,0,0,5,391,6405,46512,168245,246675,
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• V. Pilaud, J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440, 2013.

Crossrefs

Cf. A232222 (row sums), A000699 (column sums), A322456 (transpose).

Cf. A067311.

Programs

PARI

\\ M(n,m) is the n-th row of A067311 truncated at m.
M(n,m)={1/(1-y+O(y*y^m))^n*sum(k=0, (sqrtint(m*8+1)-1)\2, (-1)^k * ( binomial(2*n, n-k)-binomial(2*n, n-k-1)) * y^(k*(k+1)/2) )}
T(n)={my(g=sum(k=0,n+1,M(k,n)*x^k,O(x^2*x^n)), v=Vec(sqrt((x/serreverse( x*g^2 ))))); vector(n, n, vector(n, k, polcoef(v[2+k],n)))} \\ Andrew Howroyd, Nov 22 2024

Extensions

3 more rows. - R. J. Mathar, Dec 09 2018