A268041 Table array: T(n,m) is the number of non-crossings matchings of curves embedded within an annulus with n exterior endpoints and m interior endpoints.

1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 2, 1, 2, 0, 0, 0, 0, 0, 0, 4, 2, 3, 2, 3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 7, 3, 7, 3, 7, 5, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 14, 17, 8, 14, 8, 14, 8, 17, 14, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 80, 42, 48, 24, 38, 20, 34, 20, 38, 24, 48, 42, 80

Offset

0,11

Links

Table of n, a(n) for n=0..90.

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

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)));
a(n, m) = {if ((n+m) % 2, return (0)); if (n<m, return (a(m, n))); sum(k=0, m, if (!((n-k)%2) && !((m-k)%2), tnmk((n-k)/2, (m-k)/2, k), 0)); }

Crossrefs

Cf. A267998.

Sequence in context: A140324 A323284 A010250 * A348248 A060024 A143668

Adjacent sequences: A268038 A268039 A268040 * A268042 A268043 A268044

Keyword

nonn,tabl

Author

Michel Marcus, Jan 25 2016

Status

approved