A303870 Number of noncrossing partitions up to rotation and reflection composed of n blocks of size 4.

1, 1, 1, 3, 8, 34, 169, 1019, 6710, 47104, 342772, 2566209, 19621256, 152669854, 1205358482, 9636786366, 77890590994, 635628049370, 5231328157060, 43382605871299, 362225044991368, 3043083681629249, 25708398651274529, 218296978274674435, 1862280135781609982

Offset: 0

Author

Andrew Howroyd, May 01 2018

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Column k=4 of A303929.

Cf. A054362.

Programs

Mathematica

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #] &] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#] &])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
a[n_] := T[n, 4];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 14 2018, after _Andrew Howroyd and A303929 *)

Formula

a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jun 01 2022