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

%I A303870 #15 Jun 01 2022 05:50:52
%S A303870 1,1,1,3,8,34,169,1019,6710,47104,342772,2566209,19621256,152669854,
%T A303870 1205358482,9636786366,77890590994,635628049370,5231328157060,
%U A303870 43382605871299,362225044991368,3043083681629249,25708398651274529,218296978274674435,1862280135781609982
%N A303870 Number of noncrossing partitions up to rotation and reflection composed of n blocks of size 4.
%H A303870 Andrew Howroyd, <a href="/A303870/b303870.txt">Table of n, a(n) for n = 0..200</a>
%F A303870 a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jun 01 2022
%t A303870 u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
%t A303870 e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
%t A303870 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 A303870 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;
%t A303870 a[n_] := T[n, 4];
%t A303870 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd and A303929 *)
%Y A303870 Column k=4 of A303929.
%Y A303870 Cf. A054362.
%K A303870 nonn
%O A303870 0,4
%A A303870 _Andrew Howroyd_, May 01 2018