A300975 -id:A300975 - cp's OEIS frontend

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

1, 1, 3, 10, 39, 151, 588, 2304, 9111, 36307, 145553, 586246, 2370264, 9614242, 39105580, 159444160, 651468967, 2666771488, 10934393619, 44899828056, 184616878289, 760010818689, 3132147583744, 12921037206764, 53351800567200, 220478125956426, 911839751015196, 3773836780169050

Offset: 0

Ilya Gutkovskiy, Mar 17 2018

nonn

Number of partitions of n into squares of n kinds.

Cf. A000290, A001156, A008485, A023871, A240944, A279225, A285047, A300975, A301518.

a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^2*j, i-1), j=0..n/i^2)))
      end: b(n, isqrt(n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018

Table[SeriesCoefficient[Product[1/(1 - x^k^2)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

From Vaclav Kotesovec, Mar 23 2018: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 4.216358447600641565890184638418336163396695730036... and
c = 0.26442245016754864773722176155288663999776... (End)

A301742 a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^3)).

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777224, 387420651, 10000003000, 285311729175, 8916101692416, 302875135553107, 11112007563452544, 437893910883984375, 18446744692184842496, 827240282046275783406, 39346408782249049076832, 1978419682220092642678901, 104857601064960000960000000

Offset: 0

Ilya Gutkovskiy, Mar 26 2018

nonn

Number of compositions (ordered partitions) of n into cubes of n kinds.

Cf. A000578, A023358, A300975, A301335.

Table[SeriesCoefficient[1/(1 - n Sum[x^k^3, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

A303168 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 10, 10, 10, 13, 13, 13, 13, 13, 18, 18, 18, 24, 24, 24, 24, 24, 30, 30, 30, 39, 39, 39, 39, 39, 46, 46, 46, 58, 58, 58, 64, 64, 72, 72, 72, 87, 87, 87, 99, 99, 112, 112, 112, 130, 130, 130, 148, 148, 166, 166, 166, 187

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Number of partitions of n into 1 kind of part 1, 2 kinds of part 8, 3 kinds of part 27, ..., k kinds of part k^3.

Cf. A000219, A000578, A003108, A023872, A285047, A291720, A298434, A300975.

nmax = 75; CoefficientList[Series[Product[1/(1 - x^k^3)^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A000578(k))^k.

cp's OEIS Frontend

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

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A301742 a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^3)).

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Mathematica

A303168 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^k.

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Mathematica

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