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Triangle read by rows: T(n,k) = number of connected matchings with n crossings and k chords, in a disk, k=2..n+1.
4

%I #14 Nov 23 2024 00:29:04

%S 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,

%T 3740,7752,0,0,0,15,546,5600,25194,43263,0,0,0,5,391,6405,46512,

%U 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

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

%H Andrew Howroyd, <a href="/A232223/b232223.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%H V. Pilaud, J. Rué, <a href="http://arxiv.org/abs/1307.6440">Analytic combinatorics of chord and hyperchord diagrams with k crossings</a>, arXiv preprint arXiv:1307.6440, 2013.

%e Triangle begins:

%e 1,

%e 0,3,

%e 0,1,12,

%e 0,0,10,55,

%e 0,0,4,77,273,

%e 0,0,1,60,546,1428,

%e 0,0,0,35,624,3740,7752,

%e 0,0,0,15,546,5600,25194,43263,

%e 0,0,0,5,391,6405,46512,168245,246675,

%e ...

%o (PARI) \\ M(n,m) is the n-th row of A067311 truncated at m.

%o 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) )}

%o 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

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

%Y Cf. A067311.

%K nonn,tabl

%O 1,3

%A _N. J. A. Sloane_, Nov 22 2013

%E 3 more rows. - _R. J. Mathar_, Dec 09 2018