|
%I #10 Aug 29 2023 04:19:37
%S 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,
%T 12,30,66,132,246,1,1,5,15,43,99,217,429,810,1,1,5,19,55,143,335,715,
%U 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
%N 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.
%H Paul Drube and Puttipong Pongtanapaisan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Drube/drube3.html">Annular Non-Crossing Matchings</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.4.
%t 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 *)
%o (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));
%K nonn,tabl
%O 0,6
%A _Michel Marcus_, Jan 24 2016
|
