A279329 Number of partitions of n into distinct cubes.

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

Offset

0

Comments

In general, if m > 0 and g.f. = Product_{k>=1} (1 + x^(k^m)), then a(n) ~ exp((m+1) * ((2^(1/m)-1) * Gamma(1/m) * Zeta(1+1/m) / m^2)^(m/(m+1)) * (n/2)^(1/(m+1))) * ((2^(1/m)-1) * Gamma(1/m) * Zeta(1+1/m))^(m/(2*(m+1))) / (sqrt((m+1)*Pi) * 2^((2*m+3)/(2*(m+1))) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))).

a(12758) = 0 is the last zero in this sequence. - Antti Karttunen, Aug 30 2017

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..100000

Index entries for sequences related to sums of cubes

Index entries for related partition-counting sequences

Formula

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

a(n) ~ exp(2^(7/4) * 3^(-3/2) * ((2^(1/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4)) * ((2^(1/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/8) / (2^(17/8) * 3^(1/4) * sqrt(Pi) * n^(7/8)).

For n >= 1, a(n) = A280130(n-1) + A280130(n). - Antti Karttunen, Aug 30 2017, after Vaclav Kotesovec's Dec 26 2016 formula in the latter sequence.

Example

a(9) = 1 because we have one solution, [8, 1].
a(216) = 2 because we have two solutions: 216 = 6^3 = 5^3 + 4^3 + 3^3. This is also the first point where the sequence obtains value larger than one. - Antti Karttunen, Aug 30 2017

Mathematica

nmax = 10; CoefficientList[Series[Product[(1+x^(k^3)), {k, 1, nmax}], {x, 0, nmax^3}], x]
nmax = 10; poly = ConstantArray[0, nmax^3 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^3 + 1]], {j, nmax^3, k^3, -1}]; , {k, 2, nmax}]; poly

Prog

(PARI) A279329(n, m=1) = { my(s=0); if(!n, 1, for(c=m, n, if(ispower(c, 3), s+=A279329(n-c, c+1))); (s)); }; \\ Antti Karttunen, Aug 30 2017
(PARI) apply( A279329(n, m=1)={if(n, sum(c=m, sqrtnint(n, 3), A279329(n-c^3, c+1)), 1)}, [0..100]) \\ M. F. Hasler, Jan 05 2020
(PARI) V279329=Vecsmall(prod(k=1, sqrtnint(#l=1+O(x^N=39800), 3), l+x^k^3)-1); A279329(n)=V279329[n+!n] \\ Needs stack of N*201 byte (allocatemem) to compute the series, only (N+1)*8 byte to store the vector. - M. F. Hasler, Jan 05 2020

Crossrefs

Cf. A000009, A003108, A033461, A259792, A279226, A280130, A280263.

Cf. A001476 (positions of zeros), A003997 (positions of nonzeros after a(0)).

Cf. A030272 (a(n^3)).

Sequence in context: A281814 A353566 A279484 * A359430 A292438 A244525

Adjacent sequences: A279326 A279327 A279328 * A279330 A279331 A279332

Keyword

nonn,look

Author

Vaclav Kotesovec, Dec 10 2016

Status

approved