OFFSET
1,16
COMMENTS
In general, the number of partitions of n into perfect s-th powers (s>=1) is asymptotic to (2*Pi)^(-(s+1)/2) * sqrt(s/(s+1)) * k * n^(1/(s+1)-3/2) * exp((s+1)*k*n^(1/(s+1))), where k = (Gamma(1 + 1/s) * Zeta(1 + 1/s) / s)^(s/(s+1)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
REFERENCES
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Eric Weisstein's World of Mathematics, Partition
FORMULA
G.f.: -1+1/product(1-x^(j^4),j=1..infinity). - Emeric Deutsch, Apr 06 2006
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(1/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(13/10)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
G.f.: Sum_{i>=1} x^(i^4) / Product_{j=1..i} (1 - x^(j^4)). - Ilya Gutkovskiy, May 07 2017
EXAMPLE
a(33) = 3 because we have [16,16,1], [16,1,1,...,1] (17 1's) and [1,1,...,1] (33 1's).
MAPLE
g:=-1+1/product(1-x^(j^4), j=1..10): gser:=series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102); # Emeric Deutsch, Apr 06 2006
MATHEMATICA
g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; gser =
Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}] (* Jean-François Alcover, Oct 29 2012, after Emeric Deutsch *)
PROG
(Haskell)
a046042 = p $ tail a000583_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, May 18 2015 ~
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved