

A346793


Triangle T(n,m) read by rows: the number of n X m arrays with nonnegative integers, zeros on the border rows/columns and maximum difference one between any entry and its 4 neighbors.


0



1, 1, 2, 1, 4, 16, 1, 8, 64, 528, 1, 16, 256, 4368, 75536, 1, 32, 1024, 36176, 1312656, 48185392, 1, 64, 4096, 299664, 22844432, 1776652720, 139414770480, 1, 128, 16384, 2482384, 397748880, 65621158928, 10984586881360, 1852311458443344, 1, 256, 65536, 20563984, 6926263568
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OFFSET

1,3


COMMENTS

The arrays a(i,j) which are counted have constant surrounding borders clamped at a(i,1) = a(i,m) = a(1,j) = a(n,j) =0 , all a(i,j)>=0 and limited slopes a(i,j)a(i+1,j) <= 1, a(i,j)a(i1,j) <= 1, a(i,j)a(i,j+1) <= 1 and a(i,j)a(i,j1) <= 1. All rows and columns of the T(n,m) have rational generating functions.


LINKS



FORMULA

T(n,m) = T(m,n).
T(n,m) = 2^((m1)*(n1)), 1<=m<=3, n>=1.
T(4,m) = 9*T(4,m1) 4*T(4,m2) 16*T(4,m3).
T(5,m) = 21*T(5,m1) 52*T(5,m2) 184*T(5,m3) +32*T(5,m4) +128*T(5,m5).


EXAMPLE

The triangle starts) with n>=1, 1<=m<=n as
1
1 2
1 4 16
1 8 64 528
1 16 256 4368 75536
1 32 1024 36176 1312656 48185392
1 64 4096 299664 22844432 1776652720 139414770480
1 128 16384 2482384 397748880 65621158928 10984586881360 ...
1 256 65536 20563984 6926263568 2425367471888 867077331528016 ...


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



