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

0,3

Comments

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

Links

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.

Formula

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

Maple

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

Mathematica

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 *)

Prog

(Haskell)
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

Crossrefs

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.

Sequence in context: A369683 A261663 A199694 * A151526 A099526 A005703

Adjacent sequences: A037441 A037442 A037443 * A037445 A037446 A037447

Keyword

nonn,nice,easy

Author

Wouter Meeussen

Status

approved