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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved