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A303871 Number of noncrossing partitions up to rotation and reflection composed of n blocks of size 5.

%I A303871 #18 Jun 01 2022 05:57:14
%S A303871 1,1,1,3,11,60,423,3381,29335,266703,2507232,24177705,238003111,
%T A303871 2383370158,24217426745,249182213284,2592138293117,27225668134063,
%U A303871 288405507217589,3078471666603235,33085393411436772,357782389095170193,3890765813426578535,42527471172438573757
%N A303871 Number of noncrossing partitions up to rotation and reflection composed of n blocks of size 5.
%H A303871 Andrew Howroyd, <a href="/A303871/b303871.txt">Table of n, a(n) for n = 0..200</a>
%F A303871 a(n) ~ 5^(5*n - 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 9/2)). - _Vaclav Kotesovec_, Jun 01 2022
%t A303871 u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
%t A303871 e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
%t A303871 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 A303871 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 A303871 a[n_] := T[n, 5];
%t A303871 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ and A303929 *)
%Y A303871 Column k=5 of A303929.
%Y A303871 Cf. A054365.
%K A303871 nonn
%O A303871 0,4
%A A303871 _Andrew Howroyd_, May 01 2018