A268042 a(n) = Ann(2n) where Ann(n) is the number of annular non-crossing matchings with n total endpoints.

1, 3, 8, 20, 57, 166, 538, 1762, 6045, 21040, 74628, 267598, 970134, 3544416, 13043650, 48283236, 179665425, 671564330, 2520312810, 9492124534, 35863942748, 135893383596, 516258841134, 1965906973886, 7502329984510, 28687263026656, 109893836400756, 421684916479018

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

Formula

a(n) = Sum_{i+j=n} A268041(i, j).

Prog

(PARI) tnnk(n, k) = if (!n && !k, 1, sumdiv(gcd(n, k), d, eulerphi(d)*binomial((2*n+k)/d, n/d))/(2*n+k));
tnmk(n, m, k) = if (k==0, tnnk(n, 0)*tnnk(m, 0), k*sumdiv(gcd(k, gcd(n, m)), d, eulerphi(d)*binomial((2*n+k)/d, n/d)*binomial((2*m+k)/d, m/d))/((2*n+k)*(2*m+k)));
ann(n, m) = {if ((n+m) % 2, return (0)); if (n<m, return (ann(m, n))); sum(k=0, m, if (!((n-k)%2) && !((m-k)%2), tnmk((n-k)/2, (m-k)/2, k), 0)); }
a(n) = sum(k=0, n, ann(n-k, k));

Crossrefs

Cf. A267998, A268041.

Sequence in context: A271843 A122228 A018790 * A009436 A354519 A302674

Adjacent sequences: A268039 A268040 A268041 * A268043 A268044 A268045

Keyword

nonn

Author

Michel Marcus, Jan 25 2016

Status

approved