This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A264872 #21 Aug 14 2024 11:07:15
%S A264872 1,2,2,4,6,4,8,18,20,8,16,54,100,72,16,32,162,500,648,272,32,64,486,
%T A264872 2500,5832,4624,1056,64,128,1458,12500,52488,78608,34848,4160,128,256,
%U A264872 4374,62500,472392,1336336,1149984,270400,16512,256,512,13122,312500
%N A264872 Array read by antidiagonals: T(n,m) = 2^n*(1+2^n)^m; n,m >= 0.
%C A264872 Start with an n X m rectangle and cut it vertically along any set of the m-1 separators. There are binomial(m-1,c) ways of doing this with 0 <= c < m cuts. Inside each of these 1+c regions cut vertically, for which there are 2^(n-1) choices. The total number of ways of dissecting the rectangle into rectangles in this way is Sum_{c=0..m-1} binomial(m-1,c) 2^((1+c)(n-1)) = 2^(n-1)*(1+2^(n-1))^(m-1) = T(n-1,m-1).
%C A264872 The symmetrized version of the array is S(n,m) = T(n,m) + T(m,n) - 2^(m+n) <= A116694(n,m), which counts tilings that start with guillotine cuts either horizontally or vertically, avoiding double counting of the tilings where the order of the cuts does not matter. - _R. J. Mathar_, Nov 29 2015
%H A264872 R. J. Mathar, <a href="http://vixra.org/abs/1511.0225">Counting 2-way monotonic terrace forms over rectangular landscapes</a>, vixra:1511.0225 (2015), Section 6.
%F A264872 T(n,m) = 2^n*A264871(n,m).
%F A264872 T(n,m) <= A116694(n+1,m+1).
%e A264872 1, 2, 4, 8, 16, 32, ...
%e A264872 2, 6, 18, 54, 162, 486, ...
%e A264872 4, 20, 100, 500, 2500, 12500, ...
%e A264872 8, 72, 648, 5832, 52488, 472392, ...
%e A264872 16, 272, 4624, 78608, 1336336, 22717712, ...
%e A264872 32, 1056, 34848, 1149984, 37949472, 1252332576, ...
%e A264872 .
%e A264872 The symmetrized version S(n,m) starts
%e A264872 1, 2, 4, 8, 16, 32, ...
%e A264872 2, 8, 30, 110, 402, 1478, ...
%e A264872 4, 30, 184, 1116, 7060, 47220, ...
%e A264872 8, 110, 1116, 11600, 130968, 1622120, ...
%e A264872 16, 402, 7060, 130968, 2672416, 60666672, ...
%e A264872 32, 1478, 47220, 1622120, 60666672, 2504664128, ...
%p A264872 A264872 := proc(n,m)
%p A264872 2^n*(1+2^n)^m ;
%p A264872 end proc:
%p A264872 seq(seq(A264872(n,d-n),n=0..d),d=0..12) ; # _R. J. Mathar_, Aug 14 2024
%t A264872 Table[2^(n - m) (1 + 2^(n - m))^m, {n, 9}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 27 2015 *)
%Y A264872 Cf. A000079 (row and column 0), A008776 (row 1), A005054 (row 2), A055275 (row 3), A063376 (column 1).
%K A264872 nonn,tabl,easy
%O A264872 0,2
%A A264872 _R. J. Mathar_, Nov 27 2015