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A369575
Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(k^2)) * (1 + x^(k^3)).
2
1, 3, 4, 5, 8, 12, 16, 21, 28, 38, 51, 65, 82, 105, 133, 166, 206, 254, 312, 382, 464, 561, 677, 813, 972, 1160, 1380, 1636, 1935, 2281, 2682, 3148, 3683, 4297, 5008, 5826, 6761, 7832, 9055, 10451, 12045, 13855, 15909, 18246, 20895, 23891, 27282, 31110, 35427
OFFSET
0,2
COMMENTS
Convolution of A000009 and A033461 and A279329.
Convolution of A369570 and A279329.
a(n) is the number of triples (R(r), S(s), T(t)) where r + s + t = n, and R(k) is a partition of k into distinct parts, S(k) a partition of k into distinct squares, and T(k) a partition of k into distinct cubes.
LINKS
FORMULA
a(n) ~ exp(Pi*sqrt(n/3) + (2^(1/3) - 1) * Gamma(1/3) * zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3)) + 3^(1/4)*(sqrt(2) - 1) * zeta(3/2) * n^(1/4)/2 + 3*(2*sqrt(2) - 3) * zeta(3/2)^2 / (32*Pi)) / (8*3^(1/4)*n^(3/4)).
MATHEMATICA
nmax = 100; CoefficientList[Series[Product[(1+x^k)*(1+x^(k^2))*(1+x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jan 26 2024
STATUS
approved