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A226906 Triangle read by rows: T(n,k) is the total number of parts of size k^2, 1 <= k <= n, in the set of partitions of an n X n square lattice into squares, considering only the list of parts.

%I A226906 #24 Jan 24 2016 05:34:34
%S A226906 1,4,1,14,1,1,47,10,1,1,134,16,4,1,1,415,82,24,6,1,1,1102,165,60,16,6,
%T A226906 1,1,3076,621,169,90,22,8,1,1,7986,1361,577,194,80,28,8,1,1,20930,
%U A226906 4254,1464,643,294,114,35,10,1,1,50755,9494,3667,1491,858,297,148,41,10,1,1
%N A226906 Triangle read by rows: T(n,k) is the total number of parts of size k^2, 1 <= k <= n, in the set of partitions of an n X n square lattice into squares, considering only the list of parts.
%C A226906 The sequence was derived from the documents in the Links section.  The documents are first specified in the Links section of A034295.
%C A226906 The triangle is presented below.
%C A226906 \   k    1     2     3     4    5    6    7   8   9 10 11 12 13
%C A226906 n
%C A226906 1        1
%C A226906 2        4     1
%C A226906 3       14     1     1
%C A226906 4       47    10     1     1
%C A226906 5      134    16     4     1    1
%C A226906 6      415    82    24     6    1    1
%C A226906 7     1102   165    60    16    6    1    1
%C A226906 8     3076   621   169    90   22    8    1   1
%C A226906 9     7986  1361   577   194   80   28    8   1   1
%C A226906 10   20930  4254  1464   643  294  114   35  10   1  1
%C A226906 11   50755  9494  3667  1491  858  297  148  41  10  1  1
%C A226906 12  129977 27241 10474  4858 2239 1272  454 203  51 12  1  1
%C A226906 13  305449 60086 24702 11034 5918 2874 1474 592 249 58 12  1  1
%H A226906 Christopher Hunt Gribble, <a href="/A226906/b226906.txt">Rows n = 1..13, flattened</a>
%H A226906 Jon E. Schoenfield, <a href="https://oeis.org/A034295/a034295.txt">Table of solutions for n <= 12</a>
%H A226906 Alois P. Heinz, <a href="https://oeis.org/A034295/a034295_1.txt">More ways to divide an 11 X 11 square into sub-squares</a>
%H A226906 Alois P. Heinz, <a href="https://oeis.org/A034295/a034295_2.txt">List of different ways to divide a 13 X 13 square into sub-squares</a>
%F A226906 Sum_{k=1..n} T(n,k) * k^2 = A034295(n) * n^2.
%e A226906 For n = 3, the partitions are:
%e A226906 Square side 1 2 3
%e A226906             9 0 0
%e A226906             5 1 0
%e A226906             0 0 1
%e A226906 Total      14 1 1
%e A226906 So T(3,1) = 14, T(3,2) = 1, T(3,3) = 1.
%p A226906 b:= proc(n, l) option remember; local i, k, s, t;
%p A226906       if max(l[])>n then {} elif n=0 or l=[] then {0}
%p A226906     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p A226906     else for k do if l[k]=0 then break fi od; s:={};
%p A226906          for i from k to nops(l) while l[i]=0 do s:=s union
%p A226906              map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
%p A226906                  1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
%p A226906          od; s
%p A226906       fi
%p A226906     end:
%p A226906 T:= n-> seq(coeff(add(j, j=b(n, [0$n])), x, i), i=1..n):
%p A226906 seq(T(n), n=1..10);  # _Alois P. Heinz_, Jun 21 2013
%t A226906 b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {0}, Min[l] > 0, t = Min[l]; b[n - t, l - t], True, For[k = 1, k <= Length[l], k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[# + x^(1 + i - k)&, b[n, Join[l[[1 ;; k - 1]], Array[1 + i - k &, i - k + 1], l[[i + 1 ;; Length[l]]]]]]]; s]]; T[n_] := Table[Coefficient[Sum[j, {j, b[n, Array[0 &, n]]}], x, i], {i, 1, n}]; Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Jan 24 2016, after _Alois P. Heinz_ *)
%Y A226906 Row sums give: A226897.
%Y A226906 Cf. A034295.
%K A226906 nonn,hard,tabl
%O A226906 1,2
%A A226906 _Christopher Hunt Gribble_, Jun 21 2013