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

%I A322456 #11 Nov 23 2024 00:28:31
%S A322456 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,
%T A322456 546,273,1,7,28,84,210,462,910,1619,2618,3857,5138,6125,6405,5600,
%U A322456 3740,1428,1,8,36,120,330,792,1716,3416,6308,10872,17564,26664,38030,50864,63580,73848,78880,76296,65076,46512,25194,7752
%N A322456 Triangle of Touchard's chord enumerating polynomial coefficients [x^k] P_n(x).
%C A322456 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
%H A322456 Andrew Howroyd, <a href="/A322456/b322456.txt">Table of n, a(n) for n = 0..4090</a> (rows 0..30)
%H A322456 J. Touchard, <a href="http://dx.doi.org/10.4153/CJM-1952-001-8">Sur un problème de configurations et sur les fractions continues</a>, Canad. J. Math., 4 (1952), 2-25, P_n(x).
%F A322456 T(n,k) = [x^k] P_n(x), k=n*(n-1)/2 down to k=n-1.
%F A322456 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
%e A322456 The triangle starts
%e A322456 1;
%e A322456 1;
%e A322456 1;
%e A322456 1,  3;
%e A322456 1,  4,   10,   12;
%e A322456 1,  5,   15,   35,   60,   77,   55;
%e A322456 1,  6,   21,   56,  126,  240,  391,  546,  624,  546,  273;
%e A322456 1,  7,   28,   84,  210,  462,  910, 1619, 2618, 3857, 5138, 6125, 6405, 5600, 3740, 1428;
%o A322456 (PARI) \\ M(n) is the n-th row of A067311 as a polynomial.
%o A322456 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) )}
%o A322456 RowGfs(n)={my(g=sum(k=0,n,M(k)*x^k,O(x*x^n))); Vec(sqrt((x/serreverse( x*g^2 ))))}
%o A322456 { my(A=RowGfs(7)); for(i=1, #A, print(Vec(A[i]/y^max(0,i-2)))) } \\ _Andrew Howroyd_, Nov 22 2024
%Y A322456 Cf. A067311, A322398 (S_n(x)), A000699 (row sums), A232223 (transpose).
%K A322456 nonn,tabf
%O A322456 0,5
%A A322456 _R. J. Mathar_, Dec 09 2018
%E A322456 a(0)=1 prepended by _Andrew Howroyd_, Nov 22 2024