A267998 Table array: T(n,k) is Ann_k(2*n+k,k) where Ann_k(n,m) is the number of annular non-crossing matchings of type (n, m) with precisely k cross-cuts.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 5, 10, 1, 1, 3, 7, 14, 26, 1, 1, 4, 10, 22, 42, 80, 1, 1, 4, 12, 30, 66, 132, 246, 1, 1, 5, 15, 43, 99, 217, 429, 810, 1, 1, 5, 19, 55, 143, 335, 715, 1430, 2704, 1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32066

Offset

0,6

Links

Table of n, a(n) for n=0..77.

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

Mathematica

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#] * Binomial[(2*n+k)/#, n/#]/(2*n+k) &]; T[0, 0] = 1; Table[T[k, n-k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 29 2023 *)

Prog

(PARI) t(n, k) = if (!n && !k, 1, sumdiv(gcd(n, k), d, eulerphi(d)*binomial((2*n+k)/d, n/d))/(2*n+k));

Crossrefs

Sequence in context: A136450 A355011 A131054 * A265005 A166288 A327001

Adjacent sequences: A267995 A267996 A267997 * A267999 A268000 A268001

Keyword

nonn,tabl

Author

Michel Marcus, Jan 24 2016

Status

approved