A166771 -id:A166771 - cp's OEIS frontend

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

0, 1, 1, 2, 6, 2, 3, 15, 15, 3, 4, 28, 53, 28, 4, 5, 45, 146, 146, 45, 5, 6, 66, 356, 627, 356, 66, 6, 7, 91, 809, 2471, 2471, 809, 91, 7, 8, 120, 1759, 9292, 16213, 9292, 1759, 120, 8, 9, 153, 3716, 33878, 103196, 103196, 33878, 3716, 153, 9, 10, 190, 7702, 120771, 642364, 1123743, 642364, 120771, 7702, 190, 10

Offset: 1

Andrew Howroyd, Dec 11 2024

T(m,n) is the number of partitionings of an m X n checkerboard into two edgewise-connected sets.

			Table starts:
===================================================
m\n | 1  2    3     4      5        6         7 ...
----+----------------------------------------------
  1 | 0  1    2     3      4        5         6 ...
  2 | 1  6   15    28     45       66        91 ...
  3 | 2 15   53   146    356      809      1759 ...
  4 | 3 28  146   627   2471     9292     33878 ...
  5 | 4 45  356  2471  16213   103196    642364 ...
  6 | 5 66  809  9292 103196  1123743  12028981 ...
  7 | 6 91 1759 33878 642364 12028981 221984391 ...
  ...

Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cut.

Main diagonal is A068416.
Rows 1..4 are A001477(n-1), A000384, A378933, A378934.
Rows 3..8 multiplied by 2 are A166761, A166766, A166769, A166771, A166773, A166774.
Cf. A287151, A359990.

T(m,n) = T(n,m).

A383874 a(n) = (3n+1)!(3n)!/((2n)!*((n+1)!)^2).

Original entry on oeis.org

1, 18, 4200, 3175200, 5137292160, 14544244915200, 64008493310361600, 405192226643043840000, 3493057136053143859200000, 39378260464472988708249600000, 562659674639968187756457984000000, 9940535265182157971578474463232000000, 212816707229761791940688046273331200000000

Offset: 0

Karol A. Penson, May 22 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..150

Cf. A166149, A166771, A166494, A166384, A166750, A271049, A064352.

A383874[n_] := (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2);
Array[A383874, 15, 0] (* Paolo Xausa, May 26 2025 *)

a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); \\ Michel Marcus, May 22 2025

O.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2], (729*x)/4).
E.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2, 1], (729*x)/4).
a(n) = Integral_{x>=0} x^n*W(x)*dx, n>=0, with W(x) = MeijerG([[],[-1/2,1,1]],[[0,-1/3,-1/3,1/3,-2/3],[]],4*x/729)/(81*Pi^(3/2)), where MeijerG is the Meijer G - function. Apparently W(x) cannot be represented by any other simpler functions. W(x) is a positive function on (0,oo), is singular at x = 0 and goes monotonically to zero as x -> oo. Thus a(n) is a positive definite sequence.
  W(x) is the solution of the Stieltjes moment problem and it may be non-unique.
a(n) ~ 3^(6*n+2) * n^(2*n - 3/2) / (sqrt(Pi) * 2^(2*n+1) * exp(2*n)). - Vaclav Kotesovec, May 24 2025

cp's OEIS Frontend

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

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Formula

A383874 a(n) = (3n+1)!(3n)!/((2n)!*((n+1)!)^2).

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cp's OEIS Frontend

A378932 Array read by antidiagonals: T(m,n) is the number of minimal edge cuts in the grid graph P_m X P_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A383874 a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383874 a(n) = (3n+1)!(3n)!/((2n)!*((n+1)!)^2).