A280130 Expansion of Product_{k>=2} (1 + x^(k^3)).

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset

0

Comments

Number of partitions of n into distinct cubes > 1.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100000 (first 10001 terms from Antti Karttunen)

Index entries for sequences related to sums of cubes

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=2} (1 + x^(k^3)).

From Vaclav Kotesovec, Dec 26 2016: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k) * A279329(k).

a(n) + a(n-1) = A279329(n).

a(n) ~ A279329(n)/2.

(End)

Example

a(35) = 1 because 35 = 27 + 8. This is the first nonzero value for a noncube index.
From Antti Karttunen, Aug 30 2017: (Start)
a(72) = 1 because there is just one solution: 72 = 4^3 + 2^3.
a(216) = 2 because there are two solutions: 216 = 6^3 = 5^3 + 4^3 + 3^3. This is the first index where a(n) > 1. (End)

Mathematica

nmax = 130; CoefficientList[Series[Product[1 + x^k^3, {k, 2, nmax}], {x, 0, nmax}], x]

Prog

(PARI) A280130(n, m=2) = { my(s=0); if(!n, 1, for(c=m, n, if(ispower(c, 3), s+=A280130(n-c, c+1))); (s)); }; \\ Antti Karttunen, Aug 30 2017
(PARI)
A280130(n, m=2)={if(n, sum(c=m, sqrtnint(n, 3), A280130(n-c^3, c+1)), 1)} \\ At n ~ 2500 this is about 100 times faster than code from 2017, but for larger n (needed for A030272(n)=a(n^3)) better use (with parisize (or allocmem) >= 201*Nmax):
V280130=Vecsmall(prod(k=2, (Nmax=3*10^4)^(1/3), 1+x^k^3+O(x^Nmax))); A280130(n)=V280130[n+1] \\ M. F. Hasler, Jan 05 2020

Crossrefs

Cf. A003108, A078128, A279329.

Sequence in context: A015494 A267142 A185119 * A304002 A126811 A014057

Adjacent sequences: A280127 A280128 A280129 * A280131 A280132 A280133

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 26 2016

Status

approved