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A209612 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations and reflections.

%I A209612 #32 Jun 30 2018 02:18:32
%S A209612 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,8,8,3,1,1,3,12,17,12,3,1,1,4,19,41,
%T A209612 41,19,4,1,1,4,27,78,116,78,27,4,1,1,5,38,148,298,298,148,38,5,1,1,5,
%U A209612 50,250,680,932,680,250,50,5,1
%N A209612 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations and reflections.
%C A209612 Like the Narayana triangle A001263 (and unlike A152176) this triangle is symmetric.
%H A209612 Andrew Howroyd, <a href="/A209612/b209612.txt">Table of n, a(n) for n = 1..1275</a>
%H A209612 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rotref">Partition related number triangles</a>
%F A209612 T(n,k) = (A088855(n,k) + A209805(n,k))/2. - _Andrew Howroyd_, Nov 15 2017
%e A209612 Triangle begins:
%e A209612 1;
%e A209612 1,  1;
%e A209612 1,  1,  1;
%e A209612 1,  2,  2,  1;
%e A209612 1,  2,  4,  2,  1;
%e A209612 1,  3,  8,  8,  3,  1;
%e A209612 1,  3, 12, 17, 12,  3,  1;
%e A209612 1,  4, 19, 41, 41, 19,  4,  1;
%e A209612 1,  4, 27, 78,116, 78, 27,  4,  1;
%e A209612 1,  5, 38,148,298,298,148, 38,  5,  1
%t A209612 b[n_, k_] := Binomial[n - 1, n - k]*Binomial[n, n - k];
%t A209612 T[n_, k_] := (n*Binomial[Quotient[n - 1, 2], Quotient[k - 1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]] + DivisorSum[GCD[n, k], EulerPhi[#]* b[n/#, k/#]&] + DivisorSum[GCD[n, k - 1], EulerPhi[#]*b[n/#, (n + 1 - k)/#]&] - k*Binomial[n, k]^2/(n - k + 1))/(2*n);
%t A209612 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 30 2018, after _Andrew Howroyd_ *)
%o A209612 (PARI)
%o A209612 b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);
%o A209612 T(n,k)=(n*binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2) + sumdiv(gcd(n,k), d, eulerphi(d)*b(n/d,k/d)) + sumdiv(gcd(n,k-1), d, eulerphi(d)*b(n/d,(n+1-k)/d)) - k*binomial(n,k)^2/(n-k+1))/(2*n); \\ _Andrew Howroyd_, Nov 15 2017
%Y A209612 Cf. A111275 (row sums)
%Y A209612 Cf. A088855, A209805.
%K A209612 nonn,tabl
%O A209612 1,8
%A A209612 _Tilman Piesk_, Mar 10 2012