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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.

%I A232223 #14 Nov 23 2024 00:29:04
%S A232223 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 A232223 3740,7752,0,0,0,15,546,5600,25194,43263,0,0,0,5,391,6405,46512,
%U A232223 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 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.
%H A232223 Andrew Howroyd, <a href="/A232223/b232223.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A232223 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 A232223 Triangle begins:
%e A232223 1,
%e A232223 0,3,
%e A232223 0,1,12,
%e A232223 0,0,10,55,
%e A232223 0,0,4,77,273,
%e A232223 0,0,1,60,546,1428,
%e A232223 0,0,0,35,624,3740,7752,
%e A232223 0,0,0,15,546,5600,25194,43263,
%e A232223 0,0,0,5,391,6405,46512,168245,246675,
%e A232223 ...
%o A232223 (PARI) \\ M(n,m) is the n-th row of A067311 truncated at m.
%o A232223 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 A232223 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 A232223 Cf. A232222 (row sums), A000699 (column sums), A322456 (transpose).
%Y A232223 Cf. A067311.
%K A232223 nonn,tabl
%O A232223 1,3
%A A232223 _N. J. A. Sloane_, Nov 22 2013
%E A232223 3 more rows. - _R. J. Mathar_, Dec 09 2018