The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 remaining 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: A256557 A337471 A145179 * A232076 A329770 A099476
KEYWORD
nonn,tabl
AUTHOR
Max Alekseyev, Feb 27 2018
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 18 03:29 EDT 2024. Contains 373468 sequences. (Running on oeis4.)