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 s�ries g�n�ratrices et quelques conjectures, Dissertation, Universit� du Qu�bec � Montr�al, 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
(Python)
from functools import lru_cache
from sympy import integer_nthroot, divisors
@lru_cache(maxsize=None)
def A003108(n):
@lru_cache(maxsize=None)
def a(n): return integer_nthroot(n, 3)[1]
@lru_cache(maxsize=None)
def c(n): return sum(d for d in divisors(n, generator=True) if a(d))
return (c(n)+sum(c(k)*A003108(n-k) for k in range(1, n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved