A046042 Number of partitions of n into fourth powers.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9

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

Cf. A000583, A002377, A003105.

Cf. A001156, A003108, A046042.

Cf. A037444, A259792, A259793.

Sequence in context: A056811 A097430 A054900 * A287866 A071841 A355478

Adjacent sequences: A046039 A046040 A046041 * A046043 A046044 A046045

Keyword

nonn

Author

Eric W. Weisstein

Status

approved