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A258343
Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).
21
1, 1, 4, 14, 36, 101, 260, 669, 1669, 4116, 9932, 23636, 55483, 128532, 294422, 667026, 1496232, 3324720, 7323570, 15998749, 34679966, 74622839, 159454379, 338472749, 713956569, 1496950669, 3120663129, 6469901522, 13343153563, 27379250529, 55907749171
OFFSET
0,3
LINKS
FORMULA
a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 28 2018
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, May 28 2018
MATHEMATICA
nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)/6), {k, 1, nmax}], {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, May 27 2015
STATUS
approved