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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.
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%I #24 Jan 24 2016 05:34:34

%S 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 1,1,3076,621,169,90,22,8,1,1,7986,1361,577,194,80,28,8,1,1,20930,

%U 4254,1464,643,294,114,35,10,1,1,50755,9494,3667,1491,858,297,148,41,10,1,1

%N 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 The sequence was derived from the documents in the Links section. The documents are first specified in the Links section of A034295.

%C The triangle is presented below.

%C \ k 1 2 3 4 5 6 7 8 9 10 11 12 13

%C n

%C 1 1

%C 2 4 1

%C 3 14 1 1

%C 4 47 10 1 1

%C 5 134 16 4 1 1

%C 6 415 82 24 6 1 1

%C 7 1102 165 60 16 6 1 1

%C 8 3076 621 169 90 22 8 1 1

%C 9 7986 1361 577 194 80 28 8 1 1

%C 10 20930 4254 1464 643 294 114 35 10 1 1

%C 11 50755 9494 3667 1491 858 297 148 41 10 1 1

%C 12 129977 27241 10474 4858 2239 1272 454 203 51 12 1 1

%C 13 305449 60086 24702 11034 5918 2874 1474 592 249 58 12 1 1

%H Christopher Hunt Gribble, <a href="/A226906/b226906.txt">Rows n = 1..13, flattened</a>

%H Jon E. Schoenfield, <a href="https://oeis.org/A034295/a034295.txt">Table of solutions for n <= 12</a>

%H 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 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 Sum_{k=1..n} T(n,k) * k^2 = A034295(n) * n^2.

%e For n = 3, the partitions are:

%e Square side 1 2 3

%e 9 0 0

%e 5 1 0

%e 0 0 1

%e Total 14 1 1

%e So T(3,1) = 14, T(3,2) = 1, T(3,3) = 1.

%p b:= proc(n, l) option remember; local i, k, s, t;

%p if max(l[])>n then {} elif n=0 or l=[] then {0}

%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:={};

%p for i from k to nops(l) while l[i]=0 do s:=s union

%p map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,

%p 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))

%p od; s

%p fi

%p end:

%p T:= n-> seq(coeff(add(j, j=b(n, [0$n])), x, i), i=1..n):

%p seq(T(n), n=1..10); # _Alois P. Heinz_, Jun 21 2013

%t 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 Row sums give: A226897.

%Y Cf. A034295.

%K nonn,hard,tabl

%O 1,2

%A _Christopher Hunt Gribble_, Jun 21 2013