A346793 - OEIS

%I #9 Aug 06 2021 09:13:57

%S 1,1,2,1,4,16,1,8,64,528,1,16,256,4368,75536,1,32,1024,36176,1312656,

%T 48185392,1,64,4096,299664,22844432,1776652720,139414770480,1,128,

%U 16384,2482384,397748880,65621158928,10984586881360,1852311458443344,1,256,65536,20563984,6926263568

%N 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.

%C 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(i-1,j)| <= 1, |a(i,j)-a(i,j+1)| <= 1 and |a(i,j)-a(i,j-1)| <= 1. All rows and columns of the T(n,m) have rational generating functions.

%H R. J. Mathar, <a href="https://vixra.org/abs/2009.0152">Motzkin Islands: a 3-dimensional embeddng of Motzkin paths</a>, vixra:2009.0152 (2020) Table 2.

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

%F T(n,m) = 2^((m-1)*(n-1)), 1<=m<=3, n>=1.

%F T(4,m) = 9*T(4,m-1) -4*T(4,m-2) -16*T(4,m-3).

%F T(5,m) = 21*T(5,m-1) -52*T(5,m-2) -184*T(5,m-3) +32*T(5,m-4) +128*T(5,m-5).

%e The triangle starts) with n>=1, 1<=m<=n as

%e   1

%e   1    2

%e   1    4     16

%e   1    8     64       528

%e   1   16    256      4368        75536

%e   1   32   1024     36176      1312656       48185392

%e   1   64   4096    299664     22844432     1776652720     139414770480

%e   1  128  16384   2482384    397748880    65621158928   10984586881360 ...

%e   1  256  65536  20563984   6926263568  2425367471888  867077331528016 ...

%K nonn,tabl

%O 1,3

%A _R. J. Mathar_, Aug 04 2021