A298848 -id:A298848 - cp's OEIS frontend

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

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)).

A331899 Number of compositions (ordered partitions) of n^3 into distinct cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 127, 1, 1, 127, 769, 10945, 15961, 86641, 86521, 430717, 4140367, 4146751, 93669001, 1538834041, 663998665, 6883029151, 1014140647, 20591858857, 121532206567, 1637261351983, 2981530899847, 5950338797191, 47072230385425

Offset: 0

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(6) = 7 because we have [216], [125, 64, 27], [125, 27, 64], [64, 125, 27], [64, 27, 125], [27, 125, 64] and [27, 64, 125].

Cf. A000578, A030272, A259792, A290247, A298641, A298672, A298848, A331845, A331884.

b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^3, n, 0):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 31 2020

b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n - i^3, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^3, n, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

a(n) = A331845(A000578(n)).

cp's OEIS Frontend

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

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