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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A293426 Expansion of Product_{k>0} ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3). 3

%I #21 Oct 10 2017 10:31:19

%S 1,5,23,77,244,677,1794,4411,10454,23597,51699,109378,225804,453893,

%T 893872,1723286,3265023,6078557,11148496,20146561,35935772,63287458,

%U 110186562,189715530,323335946,545666040,912512366,1512613763,2486819428,4056167621,6566647376

%N Expansion of Product_{k>0} ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3).

%H Seiichi Manyama, <a href="/A293426/b293426.txt">Table of n, a(n) for n = 0..10000</a>

%H W. Y. C. Chen, K. Q. Ji, H.-T. Jin and E. Y. Y. Shen, <a href="https://doi.org/10.1016/j.jnt.2013.02.010">On the number of partitions with designated summands</a>, J. Number Theory, 133 (2013), 2929-2938.

%F a(n) = (1/3) * A077285(3*n+2).

%F a(n) ~ 5^(3/4) * exp(sqrt(10*n/3)*Pi) / (2^(11/4) * 3^(15/4) * n^(5/4)). - _Vaclav Kotesovec_, Oct 09 2017

%t nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k))^3 * (1 - x^(6*k))^3) / ((1 - x^k)^5 * (1 - x^(2*k))^3), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 09 2017 *)

%t max = 30; QP = QPochhammer; s = QP[q^6]/(QP[q]*QP[q^2]*QP[q^3]) + O[q]^(3 max + 3); (1/3)*Table[CoefficientList[s, q][[3*n + 3]], {n, 0, max}] (* _Jean-François Alcover_, Oct 10 2017, from 1st formula *)

%o (PARI) m = 40; Vec(prod(k=1, m, ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3)) + O(q^m)) \\ _Michel Marcus_, Oct 10 2017

%Y Cf. A077285 (PD(n)).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 09 2017

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 March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)