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%I #31 Oct 09 2023 04:34:18
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,2,2,5,1,1,1,1,1,3,3,8,
%T 1,1,1,1,2,1,9,4,13,1,1,1,1,1,4,1,16,6,21,1,1,1,1,2,1,7,2,35,9,34,1,1,
%U 1,1,1,3,1,13,3,65,13,55,1,1,1,1,2,2,9,1,46,4,143,19,89,1,1
%N Number A(n,k) of tilings of a k X n rectangle using integer-sided rectangular tiles of area k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Row n gives: 1 followed by period A003418(n): (1, A000045(n+1), ...) repeated; offset 0.
%H Alois P. Heinz, <a href="/A220122/b220122.txt">Antidiagonals n = 0..32, flattened</a>
%F For prime p column p has g.f.: 1/(1-x-x^p) or a_p(n) = Sum_{j=0..floor(n/p)} C(n-(p-1)*j,j).
%e A(4,4) = 9, because there are 9 tilings of a 4 X 4 rectangle using integer-sided rectangular tiles of area 4:
%e ._._._._. ._______. .___.___. ._.___._. ._______.
%e | | | | | |_______| | | | | | | | |_______|
%e | | | | | |_______| |___|___| | |___| | | | |
%e | | | | | |_______| | | | | | | | |___|___|
%e |_|_|_|_| |_______| |___|___| |_|___|_| |_______|
%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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
%e 1, 1, 3, 2, 3, 1, 4, 1, 3, 2, 3, ...
%e 1, 1, 5, 3, 9, 1, 7, 1, 9, 3, 5, ...
%e 1, 1, 8, 4, 16, 2, 13, 1, 16, 4, 9, ...
%e 1, 1, 13, 6, 35, 3, 46, 1, 35, 6, 15, ...
%e 1, 1, 21, 9, 65, 4, 88, 2, 65, 9, 26, ...
%e 1, 1, 34, 13, 143, 5, 209, 3, 250, 13, 44, ...
%e 1, 1, 55, 19, 281, 6, 473, 4, 495, 37, 75, ...
%e 1, 1, 89, 28, 590, 8, 1002, 5, 1209, 64, 254, ...
%p b:= proc(n, l) option remember; local i, k, m, 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, m:=0, nops(l);
%p for i from k to m while l[i]=0 do if irem(m, 1+i-k, 'q')=0
%p and q<=n then s:= s+ b(n, [l[j]$j=1..k-1, q$j=k..i,
%p l[j]$j=i+1..m]) fi od; s
%p fi
%p end:
%p A:= (n, k)-> b(n, [0$k]):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[n_, l_] := b[n, l] = Module[{i, k, m, 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, m} = {0, Length[l]}; For[ i = k , i <= m && l[[i]] == 0, i++, If[Mod[m, 1+i-k ] == 0 && (q = Quotient[m, 1+i-k]) <= n, s = s+b[n, Join[ l[[1 ;; k-1]], Array[q &, i-k+1], l[[i+1 ;; m]] ]]]]; s]]; a[n_, k_] := b[n, Array[0&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 19 2013, translated from Maple *)
%Y Columns k=0+1, 2-11, 13 give: A000012, A000045(n+1), A000930, A220123, A003520, A220124, A005709, A220125, A220126, A220127, A017905(n+11), A017907(n+13).
%Y Rows n=0+1, 2-10 give: A000012, A040001, A220128, A220129, A220130, A220131, A220132, A220133, A220134, A220135.
%Y Main diagonal gives: A182106.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Dec 05 2012
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