A003108 Number of partitions of n into cubes. (Formerly M0209)

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 23, 25, 26, 26, 27, 27, 27, 28

Offset

0,9

Comments

The g.f. 1/(z+1)/(z**2+1)/(z**4+1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong.

References

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

Links

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..100000 (terms 0..1000 from T. D. Noe)

G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Eric Weisstein's World of Mathematics, Cubic Number

Eric Weisstein's World of Mathematics, Partition

Eric Weisstein's World of Mathematics, Smarandache Sequences

Formula

G.f.: 1/Product_{j>=1} (1-x^(j^3). - Emeric Deutsch, Mar 30 2006

G.f.: Sum_{n>=0} x^(n^3) / Product_{k=1..n} (1 - x^(k^3)). - Paul D. Hanna, Mar 09 2012

a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/4)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

Example

a(16) = 3 because we have [8,8], [8,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 +...
such that the g.f. A(x) satisfies the identity [Paul D. Hanna]:
A(x) = 1/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)*(1-x^125)*...)
A(x) = 1 + x/(1-x) + x^8/((1-x)*(1-x^8)) + x^27/((1-x)*(1-x^8)*(1-x^27)) + x^64/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)) +...

Maple

g:=1/product(1-x^(j^3), j=1..30): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..65); # Emeric Deutsch, Mar 30 2006

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax^(1/3)}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 60; cmax = nmax^(1/3);
s = Table[n^3, {n, cmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

Prog

(PARI) {a(n)=polcoeff(1/prod(k=1, ceil(n^(1/3)), 1-x^(k^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 09 2012 */
(PARI) {a(n)=polcoeff(1+sum(m=1, ceil(n^(1/3)), x^(m^3)/prod(k=1, m, 1-x^(k^3)+x*O(x^n))), n)} /* Paul D. Hanna, Mar 09 2012 */
(Haskell)
a003108 = p $ tail a000578_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, Oct 31 2012
(Magma) [#RestrictedPartitions(n, {d^3:d in [1..n]}): n in [0..150]]; // Marius A. Burtea, Jan 02 2019

Crossrefs

Cf. A000578, A068980, A131799, A218495, A226748, A279329, A280263.

Cf. A001156, A046042.

Cf. A037444, A259792, A259793.

Sequence in context: A104407 A054897 A261226 * A279223 A214956 A209899

Adjacent sequences: A003105 A003106 A003107 * A003109 A003110 A003111

Keyword

nonn

Author

N. J. A. Sloane, Herman P. Robinson

Status

approved