The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #13 Jul 04 2018 02:40:35
%S 1,1,1,1,1,1,1,1,1,1,1,3,6,2,1,1,4,36,38,3,1,1,10,210,960,384,6,1,1,
%T 16,1176,18680,35956,4425,14,1,1,36,6328,313664,2280910,1588192,57976,
%U 34,1,1,64,32896,4683168,111925464,323840016,77381016,807318,95,1
%N Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.
%H Andrew Howroyd, <a href="/A303864/b303864.txt">Table of n, a(n) for n = 0..1274</a>
%e Array begins:
%e =======================================================
%e n\k| 1 2 3 4 5 6
%e ---+---------------------------------------------------
%e 0 | 1 1 1 1 1 1 ...
%e 1 | 1 1 1 3 4 10 ...
%e 2 | 1 1 6 36 210 1176 ...
%e 3 | 1 2 38 960 18680 313664 ...
%e 4 | 1 3 384 35956 2280910 111925464 ...
%e 5 | 1 6 4425 1588192 323840016 46552781760 ...
%e 6 | 1 14 57976 77381016 50668922540 21346459738384 ...
%e ...
%t nmax = 10; seq[n_, k_] := Module[{p, q, h}, p = 1 + InverseSeries[ x/(k*2^If[k == 1, 0, k - 3]*(1 + x)^k) + O[x]^n, x ]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; Integrate[((p - 1)/k + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + If[OddQ[k], 0, 2^(k/2 - 2)*x*h^(k/2)] + 1];
%t Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x];
%t T[n_, k_] := col[k][[n + 1]];
%t Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jul 04 2018, after _Andrew Howroyd_ *)
%o (PARI)
%o seq(n,k)={ \\ gives gf of k'th column
%o my(p=1 + serreverse( x/(k*2^if(k==1, 0, k-3)*(1 + x)^k) + O(x*x^n) ));
%o my(h=subst(p,x,x^2+O(x*x^n)), q=x*deriv(p)/p);
%o intformal( ((p-1)/k + sum(d=2,n,eulerphi(d)*subst(q,x,x^d+O(x*x^n))))/x) + if(k%2, 0, 2^(k/2-2)*x*h^(k/2)) + 1;
%o }
%o Mat(vector(6, k, Col(seq(7, k))))
%Y Columns 2..4 are A002995(n+1), A303865, A303866.
%Y Row n=1 is A051437(k-3).
%Y Cf. A302828, A303844, A303869.
%Y Cf. A295224 (polygon dissections), A303694 (sets of cycles instead of paths).
%K nonn,tabl
%O 0,12
%A _Andrew Howroyd_, May 01 2018