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A349575 Triangle read by rows: T(w,h) (for w >= h >= 1) is the number of distinct sets of rectangles with integer sides that tile the w X h rectangle.

%I A349575 #26 Nov 25 2021 12:35:07
%S A349575 1,2,4,3,10,21,5,22,73,191,7,44,190,698,1999,11,91,507,2276,8498,
%T A349575 27016,15,172,1176,6647,29688,116593,390304,22,326,2845,18820,100160,
%U A349575 454677,1805924
%N A349575 Triangle read by rows: T(w,h) (for w >= h >= 1) is the number of distinct sets of rectangles with integer sides that tile the w X h rectangle.
%C A349575 The position and orientation of each rectangle in a tiling is irrelevant. Squares are allowed.
%C A349575 The first column is A000041 since tiling the n X 1 rectangle (ignoring the order of the tiles) is the same thing as partitioning its length n.
%e A349575 Triangle begins
%e A349575    1
%e A349575    2   4
%e A349575    3  10   21
%e A349575    5  22   73   191
%e A349575    7  44  190   698   1999
%e A349575   11  91  507  2276   8498  27016
%e A349575   15 172 1176  6647  29688 116593  390304
%e A349575   22 326 2845 18820 100160 454677 1805924 ...
%e A349575 For w=4, h=4, 3 tiles: one 2 X 1, one 3 X 2, one 4 X 2
%e A349575   aaaa
%e A349575   aaaa
%e A349575   bbbc
%e A349575   bbbc
%e A349575 For w=8, h=7, 14 tiles: one 1 X 1, one 2 X 1, one 2 X 2, six 3 X 1, two 4 X 1, one 4 X 3, one 5 X 1, one 6 X 1
%e A349575   aaaagggl
%e A349575   aaaaffkl
%e A349575   aaaaffkl
%e A349575   bbbbbbkn
%e A349575   cccccjjj
%e A349575   ddddhhhm
%e A349575   eeeeiiim
%Y A349575 Cf. A000041, A348133.
%K A349575 nonn,tabl,more
%O A349575 1,2
%A A349575 _Lars Blomberg_, Nov 22 2021