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

1,2

Links

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

Christopher Hunt Gribble, C++ program

Formula

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

Example

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 ...

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 *)

Crossrefs

Row sums give: A229239.

Cf. A037444.

Sequence in context: A293129 A200062 A338832 * A319039 A107873 A156290

Adjacent sequences: A229465 A229466 A229467 * A229469 A229470 A229471

Keyword

nonn,tabl

Author

Christopher Hunt Gribble, Sep 24 2013

Status

approved