A209805 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 10, 10, 3, 1, 1, 3, 15, 25, 15, 3, 1, 1, 4, 26, 64, 64, 26, 4, 1, 1, 4, 38, 132, 196, 132, 38, 4, 1, 1, 5, 56, 256, 536, 536, 256, 56, 5, 1, 1, 5, 75, 450, 1260, 1764, 1260, 450, 75, 5, 1

Offset: 1

Author

Tilman Piesk, Mar 13 2012

Keywords

• nonn
• tabl

Comments

Like the Narayana triangle A001263 (and unlike A152175) this triangle is symmetric.

The diagonal entries are 1, 1, 4, 25, 196, 1764, ... which is probably sequence A001246 - the squares of the Catalan numbers.

The above conjecture about the diagonal entries T(2*n-1, n) is true since gcd(2*n-1, n) = gcd(2*n-1, n-1) = 1 and then T(2*n-1, n) simplifies to A001246(n-1) using the formula given below. - Andrew Howroyd, Nov 15 2017

Examples

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   2,   2,   1;
  1,   2,   4,   2,   1;
  1,   3,  10,  10,   3,   1;
  1,   3,  15,  25,  15,   3,   1;
  1,   4,  26,  64,  64,  26,   4,   1;
  1,   4,  38, 132, 196, 132,  38,   4,   1;
  1,   5,  56, 256, 536, 536, 256,  56,   5,   1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• Tilman Piesk, Partition related number triangles (Wikiversity)
• Tilman Piesk, Brute force MATLAB code used to calculate the rows (Pastebin)

Crossrefs

Cf. A054357 (row sums), A001246 (square Catalan numbers).

Cf. A001263, A103371, A132812, A209612.

Programs

Mathematica

b[n_, k_] := Binomial[n-1, n-k] Binomial[n, n-k];
T[n_, k_] := (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))/n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

PARI

b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);
T(n,k)=(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))/n; \\ Andrew Howroyd, Nov 15 2017

Formula

T(n,k) = (1/n)*((Sum_{d|gcd(n,k)} phi(d)*A103371(n/d-1,k/d-1)) + (Sum_{d|gcd(n,k-1)} phi(d)*A103371(n/d-1,(n+1-k)/d-1)) - A132812(n,k)). - Andrew Howroyd, Nov 15 2017