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A248884
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Expansion of Product_{k>=1} (1+x^k)^(k^5).
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9
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1, 1, 32, 275, 1763, 12421, 85808, 561074, 3535678, 21815897, 131733641, 778099521, 4505634324, 25635135074, 143507764032, 791243636305, 4300983535471, 23070300486656, 122213931799869, 639848848696540, 3312824859756453, 16972058378914997, 86082216143323410
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OFFSET
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0,3
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COMMENTS
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In general, for m > 0, if g.f. = Product_{k>=1} (1+x^k)^(k^m), then a(n) ~ 2^(zeta(-m)) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((1-2^(-m-1)) * Gamma(m+2) * zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2))) / (sqrt(2*Pi*(m+2)) * n^((m+3)/(2*m+4))).
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.
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FORMULA
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a(n) ~ (5*zeta(7))^(1/14) * 3^(2/7) * exp(zeta(7)^(1/7) * 2^(-9/7) * 3^(-3/7) * 5^(1/7) * 7^(8/7) * n^(6/7)) / (2^(163/252) * 7^(3/7) * sqrt(Pi) * n^(4/7)), where zeta(7) = A013665.
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MAPLE
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b:= proc(n) option remember; add(
(-1)^(n/d+1)*d^6, d=numtheory[divisors](n))
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(b(k)*a(n-k), k=1..n)/n)
end:
seq(a(n), n=0..35); # Alois P. Heinz, Oct 16 2017
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MATHEMATICA
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nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5), {k, 1, nmax}], {x, 0, nmax}], x]
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PROG
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(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^k^5)) \\ G. C. Greubel, Oct 31 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 3012018
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CROSSREFS
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Cf. A026007 (m=1), A027998 (m=2), A248882 (m=3), A248883 (m=4).
Column k=5 of A284992.
Sequence in context: A125369 A126527 A265842 * A223904 A122103 A009526
Adjacent sequences: A248881 A248882 A248883 * A248885 A248886 A248887
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KEYWORD
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nonn
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AUTHOR
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Vaclav Kotesovec, Mar 05 2015
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STATUS
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approved
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