The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A261389 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3*k). 6
 1, 6, 30, 128, 486, 1704, 5604, 17484, 52206, 150118, 417696, 1128984, 2973476, 7650720, 19272432, 47616568, 115570014, 275921460, 648771802, 1503889488, 3439990344, 7770915816, 17349229908, 38306180052, 83694778556, 181052778078, 387976101432, 823939048560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Convolution of A255610 and A027346. In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(t*k) and t>=1, then a(n) ~ exp(t/12 + 3/2 * (7*t*Zeta(3)/2)^(1/3) * n^(2/3)) * t^(1/6 + t/36) * (7*Zeta(3))^(1/6 + t/36) / (A^t * 2^(2/3 + t/9) * sqrt(3*Pi) * n^(2/3 + t/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. LINKS G. C. Greubel, 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. 19. FORMULA a(n) ~ exp(1/4 + 3/2 * (21*Zeta(3)/2)^(1/3) * n^(2/3)) * (7*Zeta(3)/3)^(1/4) / (2 * A^3 * sqrt(Pi) * n^(3/4)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. MATHEMATICA nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] CROSSREFS Cf. A156616 (t=1), A261386 (t=2). Cf. A015128, A027346, A027906, A193427, A255610. Sequence in context: A131458 A032205 A007465 * A073389 A320744 A232061 Adjacent sequences:  A261386 A261387 A261388 * A261390 A261391 A261392 KEYWORD nonn AUTHOR Vaclav Kotesovec, Aug 17 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 13 10:21 EDT 2021. Contains 344985 sequences. (Running on oeis4.)