A302828 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation and reflection with each path having exactly k nodes.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 21, 22, 3, 1, 1, 6, 111, 494, 201, 6, 1, 1, 10, 604, 9400, 18086, 2244, 12, 1, 1, 20, 3196, 157040, 1141055, 794696, 29096, 27, 1, 1, 36, 16528, 2342480, 55967596, 161927208, 38695548, 404064, 65, 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 2 3 6 ... 2 | 1 1 4 21 111 604 ... 3 | 1 2 22 494 9400 157040 ... 4 | 1 3 201 18086 1141055 55967596 ... 5 | 1 6 2244 794696 161927208 23276467936 ... 6 | 1 12 29096 38695548 25334545270 10673231900808 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1274
- Math StackExchange, Question from user Matan at math.stackexchange.com: Number of ways to connect sets of k dots in a perfect n-gon
Crossrefs
Programs
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Mathematica
nmax = 10; seq[n_, k_] := Module[{p, q, h, c}, p = 1 + InverseSeries[ x/(k*2^(k - 3)*(1 + x)^k) + O[x]^n, x]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; c = 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)]; If[k == 1, 2/(1 - x) + O[x]^n, 3/2 + c + If[OddQ[k], h + x^2*2^(k - 3)*h^k + x*2^((k - 1)/2)*h^((k + 1)/2), If[k == 2, x*h, 0] + h/(1 - 2^(k/2 - 1)*x*h^(k/2))]/2]/2]; 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^(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); my(c=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))); if(k==1, 2/(1-x) + O(x*x^n), 3/2 + c + if(k%2, h + x^2*2^(k-3)*h^k + x*2^((k-1)/2)*h^((k+1)/2), if(k==2,x*h,0) + h/(1-2^(k/2-1)*x*h^(k/2)) )/2)/2; } Mat(vector(6, k, Col(seq(7, k))))