A339368 -id:A339368 - cp's OEIS frontend

A339369 Number of partitions of n into an odd number of cubes.

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 4, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 6, 4, 6, 5, 6, 5, 7, 6, 7, 6, 8, 6, 8, 7, 8, 8, 9, 8, 9, 9, 9, 9, 10, 10, 11, 10, 12, 10, 12, 11, 12, 12, 14, 12, 14, 13, 14

Offset: 0

Ilya Gutkovskiy, Dec 01 2020

nonn

			a(17) = 2 because we have [8, 8, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000578, A003108, A027193, A292560, A339365, A339368.

nmax = 85; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}] - Product[1/(1 + x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k^3)) - Product_{k>=1} 1 / (1 + x^(k^3))).

a(n) = (A003108(n) - A292560(n)) / 2.

A339420 Number of compositions (ordered partitions) of n into an even number of cubes.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 2, 10, 7, 12, 16, 14, 29, 16, 46, 22, 67, 40, 94, 78, 125, 144, 161, 246, 214, 394, 312, 602, 499, 878, 835, 1236, 1396, 1722, 2286, 2446, 3637, 3614, 5598, 5560, 8358, 8782, 12226, 14014, 17776, 22278, 26056, 34924

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(11) = 4 because we have [8, 1, 1, 1], [1, 8, 1, 1], [1, 1, 8, 1] and [1, 1, 1, 8].

Cf. A000578, A023358, A034008, A323633, A339368, A339418, A339421.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020

nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) + 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) + 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) + A323633(n)) / 2.
a(n) = Sum_{k=0..n} A023358(k) * A323633(n-k).

cp's OEIS Frontend

A339369 Number of partitions of n into an odd number of cubes.

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