A303864 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.
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, 16, 1176, 18680, 35956, 4425, 14, 1, 1, 36, 6328, 313664, 2280910, 1588192, 57976, 34, 1, 1, 64, 32896, 4683168, 111925464, 323840016, 77381016, 807318, 95, 1
Offset: 0
Examples
Array begins: ======================================================= n\k| 1 2 3 4 5 6 ---+--------------------------------------------------- 0 | 1 1 1 1 1 1 ... 1 | 1 1 1 3 4 10 ... 2 | 1 1 6 36 210 1176 ... 3 | 1 2 38 960 18680 313664 ... 4 | 1 3 384 35956 2280910 111925464 ... 5 | 1 6 4425 1588192 323840016 46552781760 ... 6 | 1 14 57976 77381016 50668922540 21346459738384 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1274
Crossrefs
Programs
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Mathematica
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]; Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x]; T[n_, k_] := col[k][[n + 1]]; Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 04 2018, after Andrew Howroyd *) -
PARI
seq(n,k)={ \\ gives gf of k'th column my(p=1 + serreverse( x/(k*2^if(k==1, 0, k-3)*(1 + x)^k) + O(x*x^n) )); my(h=subst(p,x,x^2+O(x*x^n)), q=x*deriv(p)/p); 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; } Mat(vector(6, k, Col(seq(7, k))))