A229239 -id:A229239 - cp's OEIS frontend

A037444 Number of partitions of n^2 into squares.

1, 1, 2, 4, 8, 19, 43, 98, 220, 504, 1116, 2468, 5368, 11592, 24694, 52170, 108963, 225644, 462865, 941528, 1899244, 3801227, 7550473, 14889455, 29159061, 56722410, 109637563, 210605770, 402165159, 763549779, 1441686280, 2707535748, 5058654069, 9404116777

Offset: 0

Wouter Meeussen

Is lim_{n->inf} a(n)^(1/n) > 1? - Paul D. Hanna, Aug 20 2002

The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - Vaclav Kotesovec, Dec 29 2016

T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..945 (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Entries with square index in A001156.
Cf. A072964, A030273, A000041, A000290, A229239, A229468.
Cf. A003108, A046042.
Cf. A259792, A259793.
A row or column of the array in A259799.

a037444 n = p (map (^ 2) [1..]) (n^2) where
   p _      0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 15 2013

max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Oct 18 2011 *)

a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).

a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - Vaclav Kotesovec, Dec 29 2016

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.

Original entry on oeis.org

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

Christopher Hunt Gribble, Sep 24 2013

			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 ...

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

Row sums give: A229239.

Cf. A037444.

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

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 *)

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

cp's OEIS Frontend

A037444 Number of partitions of n^2 into squares.

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