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%I #24 Sep 14 2021 06:57:29
%S 1,1,1,1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0,
%T 0,1,1,0,1,0,2,0,1,0,1,1,0,0,1,0,0,1,0,0,1,1,0,1,0,3,1,3,0,1,0,1,1,0,
%U 0,0,0,2,2,0,0,0,0,1,1,0,1,1,5,0,7,0,5,1,1,0,1,1,0,0,0,0,0,7,7,0,0,0,0,0,1
%N Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A226206/b226206.txt">Antidiagonals n = 0..34, flattened</a>
%e A(6,4) = A(4,6) = 3:
%e ._._._._._._. ._._._._._._. ._._._._._._.
%e | | | | | | | | | |
%e |___|___|___| | |___| |___| |
%e | | | | | | | | | |
%e |___|___|___| |_______|___| |___|_______| .
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
%e 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, ...
%e 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, ...
%e 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, ...
%e 1, 0, 1, 1, 3, 2, 7, 7, 16, 19, 40, ...
%e 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, ...
%e 1, 0, 1, 0, 5, 0, 16, 0, 48, 0, 160, ...
%e 1, 0, 0, 1, 0, 0, 19, 0, 0, 50, 17, ...
%e 1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ...
%e ...
%p b:= proc(n, l) option remember; local i, k, s, t;
%p if max(l[])>n then 0 elif n=0 or l=[] then 1
%p elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p else for k do if l[k]=0 then break fi od; s:=0;
%p for i from k+1 to nops(l) while l[i]=0 do s:=s+
%p b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
%p od; s
%p fi
%p end:
%p A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; a [n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 11 2013, translated from Maple *)
%Y Columns (or rows) k=0-12 give: A000012, A000007, A059841, A079978, A079977, A226369, A226370, A226371, A226372, A226373, A226374, A226375, A226376.
%Y Main diagonal gives A347800.
%Y Cf. A219924.
%K nonn,tabl
%O 0,41
%A _Alois P. Heinz_, May 31 2013
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