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

1,8

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Tilman Piesk, Partition related number triangles

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A111275 (row sums)

Cf. A088855, A209805.

Sequence in context: A172479 A339788 A122085 * A209805 A238453 A066287

Adjacent sequences: A209609 A209610 A209611 * A209613 A209614 A209615

Keyword

nonn,tabl

Author

Tilman Piesk, Mar 10 2012

Status

approved