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

0,4

Links

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

Formula

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

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 *)

Crossrefs

Column k=4 of A303929.

Cf. A054362.

Sequence in context: A117722 A231856 A024419 * A186517 A094448 A063805

Adjacent sequences: A303867 A303868 A303869 * A303871 A303872 A303873

Keyword

nonn

Author

Andrew Howroyd, May 01 2018

Status

approved