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A295198 Number of noncrossing partitions up to rotation of an n-set without singleton blocks.

%I A295198 #12 Jul 03 2018 08:20:40
%S A295198 1,0,1,1,2,2,5,6,15,28,67,145,368,870,2211,5549,14290,36824,96347,
%T A295198 252927,670142,1783770,4777951,12855392,34756783,94345664,257114389,
%U A295198 703150507,1929404736,5310364234,14658134277,40569137070,112566363319,313074271844,872677323283
%N A295198 Number of noncrossing partitions up to rotation of an n-set without singleton blocks.
%H A295198 Andrew Howroyd, <a href="/A295198/b295198.txt">Table of n, a(n) for n = 0..200</a>
%F A295198 a(n) = (1/n) * (A005043(n) - A002426(n) + Sum_{d|n} phi(n/d) * A002426(d)).
%t A295198 b[0] = 1; b[1] = 0; b[n_] := b[n] = (n-1)*(2*b[n-1] + 3*b[n-2])/(n+1);
%t A295198 a[0] = 1; a[n_] := (b[n] + Sum[EulerPhi[n/d]*Coefficient[(1 + x + x^2)^d, x, d], {d, Most @ Divisors[n]}])/n;
%t A295198 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 03 2018, after _Andrew Howroyd_ *)
%o A295198 (PARI)  \\ here b(n) is A005043.
%o A295198 b(n) = {polcoeff(serreverse((x - x^3) / (1 + x^3) + x * O(x*x^n)), n+1)}
%o A295198 a(n) = {if(n<1, n==0, (b(n) + sumdiv(n,d, if(d<n, eulerphi(n/d) * polcoeff((1 + x + x^2)^d, d))))/n)}
%Y A295198 Column k=0 of A211357.
%Y A295198 Cf. A005043 (noncrossing partitions of an n-set without singleton blocks).
%Y A295198 Cf. A002426.
%K A295198 nonn
%O A295198 0,5
%A A295198 _Andrew Howroyd_, Nov 16 2017