A211357 Triangle read by rows: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that contain k singleton blocks.

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 3, 2, 2, 0, 1, 5, 6, 9, 4, 3, 0, 1, 6, 15, 18, 15, 5, 3, 0, 1, 15, 36, 56, 42, 29, 7, 4, 0, 1, 28, 91, 144, 142, 84, 42, 10, 4, 0, 1, 67, 232, 419, 432, 322, 152, 66, 12, 5, 0, 1, 145, 603, 1160, 1365, 1080, 630, 252, 90, 15, 5, 0, 1

Offset: 0

Author

Tilman Piesk, Apr 12 2012

Keywords

• nonn
• tabl

Examples

			From _Andrew Howroyd_, Nov 16 2017: (Start)
Triangle begins: (n >= 0, 0 <= k <= n)
   1;
   0,   1;
   1,   0,   1;
   1,   1,   0,   1;
   2,   1,   2,   0,   1;
   2,   3,   2,   2,   0,   1;
   5,   6,   9,   4,   3,   0,  1;
   6,  15,  18,  15,   5,   3,  0,  1;
  15,  36,  56,  42,  29,   7,  4,  0, 1;
  28,  91, 144, 142,  84,  42, 10,  4, 0, 1;
  67, 232, 419, 432, 322, 152, 66, 12, 5, 0, 1;
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..90 from Tilman Piesk)

Crossrefs

Column k=0 is A295198.

Row sums are A054357.

Cf. A091867 (noncrossing partitions of an n-set with k singleton blocks), A211359 (up to rotations and reflections).

Cf. A171128.

Programs

Mathematica

a91867[n_, k_] := If[k == n, 1, (Binomial[n + 1, k]/(n + 1)) Sum[Binomial[n + 1 - k, j] Binomial[n - k - j - 1, j - 1], {j, 1, (n - k)/2}]];
a2426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}];
a171128[n_, k_] := Binomial[n, k]*a2426[n - k];
T[0, 0] = 1;
T[n_, k_] := (1/n)*(a91867[n, k] - a171128[n, k] + Sum[EulerPhi[d]* a171128[n/d, k/d], {d, Divisors[GCD[n, k]]}]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

PARI

g(x,y) = {1/sqrt((1 - (1 + y)*x)^2 - 4*x^2) - 1}
S(n)={my(A=(1-sqrt(1-4*x/(1-(y-1)*x) + O(x^(n+2))))/(2*x)-1); Vec(1+intformal((A + sum(k=2, n, eulerphi(k)*g(x^k + O(x*x^n), y^k)))/x))}
my(v=S(10)); for(n=1, #v, my(p=v[n]); for(k=0, n-1, print1(polcoeff(p,k), ", ")); print) \\ Andrew Howroyd, Nov 16 2017

Formula

T(n,k) = (1/n)*(A091867(n,k) - A171128(n,k) + Sum_{d|gcd(n,k)} phi(d) * A171128(n/d, k/d)) for n > 0. - Andrew Howroyd, Nov 16 2017