A264392 -id:A264392 - cp's OEIS frontend

A264391 Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 6, 5, 4, 2, 2, 1, 1, 0, 1, 8, 6, 5, 4, 2, 2, 1, 1, 0, 1, 11, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1, 13, 12, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1, 19, 15, 12, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 0

Emeric Deutsch, Nov 13 2015

Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,0) = A264393(n).
Sum_{k=0..n}k*T(n,k) = A264392(n) = total number of perfect cube parts in all partitions of n.

			T(7,1) = 4 because we have [6,1],[4,2,1],[3,3,1], and [2,2,2,1] (the partitions of 7 that have 1 perfect cube part).
Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  2, 1, 1, 0, 1;
  2, 2, 1, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A264392, A264393.

h := proc (i) options operator, arrow: i^3 end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
q:= proc(n) option remember; `if`(n=iroot(n, 3)^3, 1, 0) end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+x^q(i)*b(n-i, min(i, n-i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 14 2020

cnt[P_List] := Count[P, p_ /; IntegerQ[p^(1/3)]];
cnts[n_] := cnts[n] = cnt /@ IntegerPartitions[n];
T[n_, k_] := Count[cnts[n], k];
Table[T[n, k], {n, 0, 18}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2020 *)

G.f.: G(t,x) = Product_{i>=1} (1-x^h(i))/((1-x^i)*(1-t*x^h(i))), where h(i) = i^3.

A342229 Total sum of parts which are cubes in all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 19, 30, 53, 75, 113, 163, 235, 328, 461, 628, 868, 1163, 1564, 2069, 2743, 3578, 4674, 6036, 7795, 9962, 12728, 16151, 20441, 25714, 32290, 40332, 50292, 62405, 77288, 95339, 117382, 143987, 176298, 215168, 262121, 318385, 386043, 466838, 563577, 678712

Offset: 0

Ilya Gutkovskiy, Mar 06 2021

nonn

			For n = 4 we have:
--------------------------------
Partitions        Sum of parts
.               which are cubes
--------------------------------
4 ................... 0
3 + 1 ............... 1
2 + 2 ............... 0
2 + 1 + 1 ........... 2
1 + 1 + 1 + 1 ....... 4
--------------------------------
Total ............... 7
So a(4) = 7.

Cf. A000041, A000578, A066186, A113061, A264392, A342228.

nmax = 45; CoefficientList[Series[Sum[k^3 x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}]/Product[(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[DivisorSum[k, # &, IntegerQ[#^(1/3)] &] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]

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

a(n) = Sum_{k=1..n} A113061(k) * A000041(n-k).

cp's OEIS Frontend

A264391 Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n).

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