A229468 Number T(n,k) of parts of each size k^2 in all partitions of n^2 into squares; triangle T(n,k), 1 <= k <= n, read by rows.

1, 4, 1, 15, 3, 1, 50, 11, 2, 1, 156, 35, 10, 4, 1, 460, 101, 36, 14, 4, 1, 1296, 298, 105, 44, 16, 6, 1, 3522, 798, 300, 130, 56, 23, 6, 1, 9255, 2154, 827, 377, 174, 82, 31, 9, 1, 23672, 5490, 2164, 1015, 502, 243, 108, 43, 10, 1, 59050, 13914, 5525, 2658, 1350, 705, 343, 154, 55, 13, 1

Offset: 1

Author

Christopher Hunt Gribble, Sep 24 2013

Keywords

• nonn
• tabl

Examples

			For n = 3, the 4 partitions are:
Square side 1 2 3
            9 0 0
            5 1 0
            1 2 0
            0 0 1
Total      15 3 1
So T(3,1) = 15, T(3,2) = 3, T(3,3) = 1.
The triangle begins:
.\ k    1     2     3     4     5     6     7     8     9 ...
.n
.1      1
.2      4     1
.3     15     3     1
.4     50    11     2     1
.5    156    35    10     4     1
.6    460   101    36    14     4     1
.7   1296   298   105    44    16     6     1
.8   3522   798   300   130    56    23     6     1
.9   9255  2154   827   377   174    82    31     9     1
10  23672  5490  2164  1015   502   243   108    43    10 ...
11  59050 13914  5525  2658  1350   705   343   154    55 ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened (Rows n = 1..21 from Christopher Hunt Gribble)
• Christopher Hunt Gribble, C++ program

Crossrefs

Row sums give: A229239.

Cf. A037444.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
      `if`(i^2>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i^2, i))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n^2, n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 24 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1+n*x, b[n, i-1] + If[i^2>n, 0, Function[ {g}, g+Coefficient[g, x, 0]*x^i][b[n-i^2, i]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][ b[n^2, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Formula

Sum_{k=1..n} T(n,k) * k^2 = A037444(n) * n^2.