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A248884 Expansion of Product_{k>=1} (1+x^k)^(k^5). 8
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<x>:=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 October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)