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 A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5). 9
 1, 1, 32, 275, 1763, 12421, 85808, 561074, 3535678, 21815897, 131733641, 778099521, 4505634324, 25635135074, 143507764032, 791243636305, 4300983535471, 23070300486656, 122213931799869, 639848848696540, 3312824859756453, 16972058378914997, 86082216143323410 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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))). LINKS 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. FORMULA 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. MAPLE 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 MATHEMATICA nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^5), {k, 1, nmax}], {x, 0, nmax}], x] PROG (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:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^5: k in [1..m]]) )); // G. C. Greubel, Oct 3012018 CROSSREFS 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 KEYWORD nonn AUTHOR Vaclav Kotesovec, Mar 05 2015 STATUS approved

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Last modified April 1 06:57 EDT 2023. Contains 361673 sequences. (Running on oeis4.)