login
This site is supported by donations 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

Table of n, a(n) for n=0..38.

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

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

License Agreements, Terms of Use, Privacy Policy. .

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