A295198 Number of noncrossing partitions up to rotation of an n-set without singleton blocks.

1, 0, 1, 1, 2, 2, 5, 6, 15, 28, 67, 145, 368, 870, 2211, 5549, 14290, 36824, 96347, 252927, 670142, 1783770, 4777951, 12855392, 34756783, 94345664, 257114389, 703150507, 1929404736, 5310364234, 14658134277, 40569137070, 112566363319, 313074271844, 872677323283

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(n) = (1/n) * (A005043(n) - A002426(n) + Sum_{d|n} phi(n/d) * A002426(d)).

Mathematica

b[0] = 1; b[1] = 0; b[n_] := b[n] = (n-1)*(2*b[n-1] + 3*b[n-2])/(n+1);
a[0] = 1; a[n_] := (b[n] + Sum[EulerPhi[n/d]*Coefficient[(1 + x + x^2)^d, x, d], {d, Most @ Divisors[n]}])/n;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

Prog

(PARI)  \\ here b(n) is A005043.
b(n) = {polcoeff(serreverse((x - x^3) / (1 + x^3) + x * O(x*x^n)), n+1)}
a(n) = {if(n<1, n==0, (b(n) + sumdiv(n, d, if(d<n, eulerphi(n/d) * polcoeff((1 + x + x^2)^d, d))))/n)}

Crossrefs

Column k=0 of A211357.

Cf. A005043 (noncrossing partitions of an n-set without singleton blocks).

Cf. A002426.

Sequence in context: A368727 A320296 A329908 * A080880 A305606 A369634

Adjacent sequences: A295195 A295196 A295197 * A295199 A295200 A295201

Keyword

nonn

Author

Andrew Howroyd, Nov 16 2017

Status

approved