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%I #25 Aug 28 2023 11:16:35
%S 1,3,15,10,60,260,35,231,1050,4375,126,882,4140,17640,72324,462,3366,
%T 16170,70070,291060,1183644,1716,12870,62920,276276,1159704,4756752,
%U 19253520,6435,49335,244530,1085175,4594590,18981270,77297220,311949495,24310,189618,950300
%N Triangular array read by row: T(m,n) = number of ways to obtain a single sphere by gluing the (labeled) sides of a (2m+1)-gon and a (2n+1)-gon, m >= n >= 0.
%C Number of ways to obtain a sphere by gluing the sides of a single 2n-gon equals Catalan number A000108(n).
%H N. Alexeev, P. Avdeyev, and M. A. Alekseyev. <a href="https://bmcbioinformatics.biomedcentral.com/track/pdf/10.1186/s12859-016-1263-7">Comparative Genomics Meets Topology: a Novel View on Genome Median and Halving Problems</a>. BMC Bioinformatics 17:Suppl 14 (2016), 3. doi:10.1186/s12859-016-1263-7
%F T(m,n) = (2*m*n+m+n+1)/(m+n+1) * binomial(2*m+1,m) * binomial(2*n+1,n).
%e Array starts:
%e m=0: 1
%e m=1: 3, 15
%e m=2: 10, 60, 260
%e m=3: 35, 231, 1050, 4375
%e m=4: 126, 882, 4140, 17640, 72324
%e m=5: 462, 3366, 16170, 70070, 291060, 1183644
%e m=6: 1716, 12870, 62920, 276276, 1159704, 4756752, 19253520
%e m=7: 6435, 49335, 244530, 1085175, 4594590, 18981270, 77297220, 311949495
%e ...
%e For m=n=1, let P and Q be triangles. They can be glued into a sphere in two manners: (1) by gluing each side of P to a side of Q, which can be done in 2*3=6 ways, where factor 2 stands for choosing orientation of gluing and factor 3 accounts for matchings of the edges across P and Q to glue with respect to the chosen orientation; or (2) by first gluing a pair of edges of P (chosen in 3 ways) together and gluing a pair of edges of Q (chosen in 3 ways) together, and then gluing the remaining single edges of P and Q, which overall can be done in 3*3=9 ways. Hence, T(1,1) = 6 + 9 = 15.
%t T[n_, k_] := With[{m=n+k+1}, 4^m (2*n*k+m) (n+1/2)! (k+1/2)!/(Pi m (n+1)! (k+1)!)];
%t Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* _Peter Luschny_, Feb 28 2018 *)
%o (PARI) { A297897(m,n) = (2*m*n+m+n+1) * binomial(2*m+1,m) * binomial(2*n+1,n) / (m+n+1); }
%Y Cf. A001700 (T(m,0)), A000108.
%K nonn,tabl
%O 0,2
%A _Max Alekseyev_, Feb 27 2018
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