OFFSET
1,2
COMMENTS
The sequence was derived from the documents in the Links section. The documents are first specified in the Links section of A034295.
The triangle is presented below.
\ k 1 2 3 4 5 6 7 8 9 10 11 12 13
n
1 1
2 4 1
3 14 1 1
4 47 10 1 1
5 134 16 4 1 1
6 415 82 24 6 1 1
7 1102 165 60 16 6 1 1
8 3076 621 169 90 22 8 1 1
9 7986 1361 577 194 80 28 8 1 1
10 20930 4254 1464 643 294 114 35 10 1 1
11 50755 9494 3667 1491 858 297 148 41 10 1 1
12 129977 27241 10474 4858 2239 1272 454 203 51 12 1 1
13 305449 60086 24702 11034 5918 2874 1474 592 249 58 12 1 1
LINKS
Christopher Hunt Gribble, Rows n = 1..13, flattened
Jon E. Schoenfield, Table of solutions for n <= 12
Alois P. Heinz, More ways to divide an 11 X 11 square into sub-squares
FORMULA
Sum_{k=1..n} T(n,k) * k^2 = A034295(n) * n^2.
EXAMPLE
For n = 3, the partitions are:
Square side 1 2 3
9 0 0
5 1 0
0 0 1
Total 14 1 1
So T(3,1) = 14, T(3,2) = 1, T(3,3) = 1.
MAPLE
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then {} elif n=0 or l=[] then {0}
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od; s:={};
for i from k to nops(l) while l[i]=0 do s:=s union
map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
od; s
fi
end:
T:= n-> seq(coeff(add(j, j=b(n, [0$n])), x, i), i=1..n):
seq(T(n), n=1..10); # Alois P. Heinz, Jun 21 2013
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
Christopher Hunt Gribble, Jun 21 2013
STATUS
approved