login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A263144 Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k. 6
1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 9, 9, 9, 9, 13, 19, 23, 23, 23, 28, 42, 51, 56, 56, 62, 84, 108, 120, 126, 133, 170, 219, 253, 268, 283, 335, 427, 503, 547, 574, 658, 815, 977, 1080, 1144, 1265, 1534, 1836, 2068, 2209, 2408, 2832, 3396, 3864, 4178, 4505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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

FORMULA

G.f.: exp(Sum_{j>=1} 1/j*x^j/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(79/900) * exp(d54 - Pi^4/(675*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (75*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(371/900) * 5^(79/450) * sqrt(3*Pi) * n^(529/900)), where d54 = A263181 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 4/(25*x^2) - 71/(300*x*exp(x)) = 0.1863826906247526303913683646299184833844240863417644... .

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

`if`(irem(d+5, 5, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..100); # after Alois P. Heinz

MATHEMATICA

nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-4))^k, {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A263141, A263142, A263143, A263181, A263148.

Sequence in context: A130497 A178154 A270774 * A126715 A158805 A163469

Adjacent sequences: A263141 A263142 A263143 * A263145 A263146 A263147

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Oct 10 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 10:26 EST 2022. Contains 358656 sequences. (Running on oeis4.)