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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Christopher Hunt Gribble and 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-Fran├žois 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

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Last modified September 28 07:51 EDT 2021. Contains 347703 sequences. (Running on oeis4.)