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

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, 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, 50, 250, 680, 932, 680, 250, 50, 5, 1

Offset: 1

Author

Tilman Piesk, Mar 10 2012

Keywords

• nonn
• tabl

Comments

Like the Narayana triangle A001263 (and unlike A152176) this triangle is symmetric.

Examples

			Triangle begins:
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, 41, 19,  4,  1;
1,  4, 27, 78,116, 78, 27,  4,  1;
1,  5, 38,148,298,298,148, 38,  5,  1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• Tilman Piesk, Partition related number triangles

Crossrefs

Cf. A111275 (row sums)

Cf. A088855, A209805.

Programs

Mathematica

b[n_, k_] := Binomial[n - 1, n - k]*Binomial[n, n - k];
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);
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

PARI

b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);
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

Formula

T(n,k) = (A088855(n,k) + A209805(n,k))/2. - Andrew Howroyd, Nov 15 2017