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 A297897 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. 1
 1, 3, 15, 10, 60, 260, 35, 231, 1050, 4375, 126, 882, 4140, 17640, 72324, 462, 3366, 16170, 70070, 291060, 1183644, 1716, 12870, 62920, 276276, 1159704, 4756752, 19253520, 6435, 49335, 244530, 1085175, 4594590, 18981270, 77297220, 311949495, 24310, 189618, 950300 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of ways to obtain a sphere by gluing the sides of a single 2n-gon equals Catalan number A000108(n). LINKS N. Alexeev, P. Avdeyev, and M. A. Alekseyev. Comparative Genomics Meets Topology: a Novel View on Genome Median and Halving Problems. BMC Bioinformatics 17:Suppl 14 (2016), 3. doi:10.1186/s12859-016-1263-7 FORMULA T(m,n) = (2*m*n+m+n+1)/(m+n+1) * binomial(2*m+1,m) * binomial(2*n+1,n). EXAMPLE Array starts: m=0:    1 m=1:    3,    15 m=2:   10,    60,    260 m=3:   35,   231,   1050,    4375 m=4:  126,   882,   4140,   17640,   72324 m=5:  462,  3366,  16170,   70070,  291060,  1183644 m=6: 1716, 12870,  62920,  276276, 1159704,  4756752, 19253520 m=7: 6435, 49335, 244530, 1085175, 4594590, 18981270, 77297220, 311949495 ... 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 remaing single edges of P and Q, which overall can be done in 3*3=9 ways. Hence, T(1,1) = 6 + 9 = 15. MATHEMATICA 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)!)]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Peter Luschny, Feb 28 2018 *) PROG (PARI) { A297897(m, n) = (2*m*n+m+n+1) * binomial(2*m+1, m) * binomial(2*n+1, n) / (m+n+1); } CROSSREFS Cf. A001700 (T(m,0)), A000108. Sequence in context: A138006 A256557 A145179 * A232076 A099476 A063628 Adjacent sequences:  A297894 A297895 A297896 * A297898 A297899 A297900 KEYWORD nonn,tabl AUTHOR Max Alekseyev, Feb 27 2018 STATUS approved

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Last modified August 24 13:48 EDT 2019. Contains 326279 sequences. (Running on oeis4.)