A030272 Number of partitions of n^3 into distinct cubes.

1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 6, 6, 7, 6, 20, 18, 21, 42, 55, 52, 80, 126, 140, 201, 323, 361, 600, 626, 938, 1387, 1648, 2310, 3620, 4575, 5495, 9278, 11239, 14229, 23406, 28780, 38218, 53987, 73114, 87568, 134007, 181986, 233004, 348230, 432184

Offset: 0

Author

Warren D. Smith

Keywords

• nonn

Examples

			a(6) = 2: [27,64,125], [216].
a(9) = 3: [1,27,64,125,512], [1,216,512], [729].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..180

Crossrefs

Cf. A000009, A003108, A033461, A259792, A279329.

Programs

Mathematica

nmax = 50; 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}]; Table[poly[[1 + n^3]], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 19 2020 *)

PARI

apply( A030272(n)=A279329(n^3), [0..30]) \\ M. F. Hasler, Jan 05 2020

Formula

a(n) = [x^(n^3)] Product_{k>=1} (1 + x^(k^3)). - Ilya Gutkovskiy, Apr 13 2017

a(n) = A279329(n^3). - Vaclav Kotesovec, May 06 2019

a(n) ~ exp(2^(7/4) * 3^(-3/2) * ((2^(1/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(3/4)) * ((2^(1/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/8) / (2^(17/8) * 3^(1/4) * sqrt(Pi) * n^(21/8)). - Vaclav Kotesovec, May 06 2019

Extensions

a(0)=1 prepended by Ilya Gutkovskiy, Apr 13 2017