A218495 -id:A218495 - cp's OEIS frontend

A003108 Number of partitions of n into cubes.

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

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

nonn

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.

			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)) +...

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.

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

Cf. A000578, A068980, A131799, A218495, A226748, A279329, A280263.
Cf. A001156, A046042.
Cf. A037444, A259792, A259793.

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

[#RestrictedPartitions(n,{d^3:d in [1..n]}): n in [0..150]]; // Marius A. Burtea, Jan 02 2019

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

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 *)

{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 */

{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 */

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

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

A218494 Number of partitions of n^3 into squares.

Original entry on oeis.org

1, 1, 3, 21, 220, 2846, 41621, 670568, 11570877, 210605770, 3998468431, 78556582448, 1587757499592, 32866068149376, 694307793698105, 14927522659021682, 325895131806047690, 7211436102222542901, 161493494674514291108, 3655277488432342084426

Offset: 0

Reinhard Zumkeller, Oct 31 2012

nonn

			n=2: number of partitions of 8 into parts of {1, 4}:
a(2) = #{4+4, 4+1+1+1+1, 8x1} = 3;
n=3: number of partitions of 27 into parts of  {1, 4, 9, 16, 25}:
a(3) = #{25+1+1, 16+9+1+1, 16+4+4+1, 16+4+5x1, 16+9x1, 9+9+9, 9+9+4+4+1, 9+9+4+5x1, 9+9+9x1, 9+4x4+1+1, 9+3x4+6x1, 9+4+4+10x1, 9+4+14x1, 9+18x1, 6x4+3x1, 5x4+7x1, 4x4+11x1, 3x4+15x1, 4+4+19x1, 4+23x1, 27x1} = 21.

Alois P. Heinz, Table of n, a(n) for n = 0..80

Cf. A000290, A001156, A218495.

a218494 = p (tail a000290_list) . (^ 3) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
       b(n, i-1)+`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^3, isqrt(n^3)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 08 2012

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i^2>n, 0, b[n - i^2, i]]]; a[n_] := b[n^3, Sqrt[n^3] // Floor]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A001156(A000578(n)).
a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/2) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n) [after Hardy & Ramanujan]. - Vaclav Kotesovec, Apr 10 2017
a(n) = [x^(n^3)] Product_{k>=1} 1/(1 - x^(k^2)). - Ilya Gutkovskiy, Jan 29 2018

Extended beyond a(7) by Alois P. Heinz, Nov 08 2012

A298934 Number of partitions of n^2 into distinct cubes.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 3, 0, 2, 4, 0, 0, 1, 0, 0, 2, 3, 1, 1, 0, 6, 3, 6, 1, 6, 0, 3, 9, 0, 6, 6, 7, 0, 10, 3, 3, 6, 0, 8, 6, 13, 2, 10, 9, 10, 19, 2, 14, 21, 7, 2, 25

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(15) = 2 because we have [216, 8, 1] and [125, 64, 27, 8, 1].

Cf. A000290, A000578, A030272, A030273, A218495, A259792, A279329, A298672, A298848, A298935.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i^2*(i+1)^2/4, 0, b(n, i-1)+
      `if`(i^3>n, 0, b(n-i^3, i-1))))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 29 2018

Table[SeriesCoefficient[Product[1 + x^k^3, {k, 1, Floor[n^(2/3) + 1]}], {x, 0, n^2}], {n, 0, 84}]

a(n) = [x^(n^2)] Product_{k>=1} (1 + x^(k^3)).

a(n) = A279329(A000290(n)).

A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 0, 6, 6, 20, 120, 7, 1689, 6636, 36540, 64020, 963996, 2894892, 19555965, 176079995, 955611188, 6684303780, 42462792168, 292378003753, 1886275214112, 13384059605364, 87399249887334, 624073002367892, 5080120229014734, 37587589611771480

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(3) = 6 because we have [8, 1, 0], [8, 0, 1], [1, 8, 0], [1, 0, 8], [0, 8, 1] and [0, 1, 8].

Index entries for sequences related to sums of cubes

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298937.

f:= n -> coeff(add(x^(k^3),k=0..floor(n^(2/3)))^n,x,n^2):
map(f, [$0..30]); # Robert Israel, Jan 29 2018

Table[SeriesCoefficient[Sum[x^k^3, {k, 0, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 0, 25}]

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^3))^n.

A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 9240, 34650, 1716, 48477, 551915, 6726720, 89973520, 102639744, 1824625081, 9915389400, 30143458884, 278196062760, 1995766236541, 6611689457736, 64547920386450, 236756174748626, 2315743488707806

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(7) = 7 because we have [8, 8, 8, 8, 8, 8, 1], [8, 8, 8, 8, 8, 1, 8], [8, 8, 8, 8, 1, 8, 8], [8, 8, 8, 1, 8, 8, 8], [8, 8, 1, 8, 8, 8, 8], [8, 1, 8, 8, 8, 8, 8] and [1, 8, 8, 8, 8, 8, 8].

Index entries for sequences related to sums of cubes

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298936.

Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 1, 27}]]

a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^3))^n.

A369087 Number of partitions of the square of the n-th triangular number into cubes.

Original entry on oeis.org

1, 1, 2, 7, 39, 289, 2514, 24625, 262768, 2998410, 36104634, 454307830, 5930683832, 79871973688, 1104858227039, 15642564963293, 226021630964469, 3325082850102429, 49705611691011214, 753760884548732440, 11578963116012317128, 179963257865062829685, 2826949471456067702388, 44840817231090333901577

Offset: 0

Paul D. Hanna, Jan 23 2024

nonn

The sum of the first n cubes equals the square of the n-th triangular number: (1^3 + 2^3 + 3^3 + ... + n^3) = (1 + 2 + 3 + ... + n)^2 = (n*(n+1)/2)^2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Cf. A000217, A000578, A003108, A000537, A218495.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> (t-> b(t, iroot(t, 3)))((n*(n+1)/2)^2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jan 24 2024

Table[SeriesCoefficient[Product[1/(1 - x^(k^3)), {k, 1, (n*(n + 1)/2)^(2/3) + 1}], {x, 0, (n*(n + 1)/2)^2}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2024 *)

{a(n) = my(M=(n*(n+1)/2)^2); polcoeff(1/prod(k=1, ceil(M^(1/3)), 1-x^(k^3)+x*O(x^M)), M)}
for(n=0,30, print1(a(n),", "))

a(n) equals the coefficient of x^A000537(n) in Product_{k>=1} 1/(1 - x^(k^3)) for n >= 0, where A000537(n) = (n*(n+1)/2)^2.
a(n) = A003108(A000537(n)) for n >= 0.
Limit a(n)^(1/n) = exp( (Gamma(1/3) * Zeta(4/3))^(3/4) * (2/3)^(3/2) ) = 19.6784468967446... - see formula by Vaclav Kotesovec in A003108.
a(n) ~ (Gamma(1/3)*zeta(4/3))^(3/4) * exp(sqrt(2) * 3^(-3/2) * (Gamma(1/3)*zeta(4/3))^(3/4) * (2*n+1)) / (3 * sqrt(2) * Pi^2 * n^5). - Vaclav Kotesovec, Jan 25 2024

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A003108 Number of partitions of n into cubes.

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A218494 Number of partitions of n^3 into squares.

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A298934 Number of partitions of n^2 into distinct cubes.

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A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

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A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes.

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A369087 Number of partitions of the square of the n-th triangular number into cubes.

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