A279762 - OEIS

%I #9 Nov 09 2017 04:46:43

%S 1,1,13,61,263,1094,4578,18076,69815,262242,965342,3480006,12322360,

%T 42896002,147062818,497000146,1657470977,5459160063,17772284155,

%U 57225458626,182362100816,575463112191,1799106136923,5575063264825,17130798464652,52216240087807,157937816918539,474197830869573,1413695306175884,4185962563381518

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k^2-5*k+2)/2).

%C Euler transform of the icosahedral numbers (A006564).

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Platonic_numbers">Platonic numbers</a>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(5*k^2-5*k+2)/2).

%F a(n) ~ exp(Zeta'(-1) + 5*Zeta(3) / (8*Pi^2) - Pi^16 / (16796160000*Zeta(5)^3) + Pi^8 * Zeta(3) / (648000*Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + 5*Zeta'(-3)/2 + (-Pi^12/(19440000 * 2^(2/5) * 15^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (21600 * 2^(4/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(4/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + (-Pi^4 / (36 * 2^(1/5) * (15*Zeta(5))^(3/5))) * n^(3/5) + ((5*(15*Zeta(5))^(1/5)) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(9/80) / (2^(11/40) * 5^(31/80) * sqrt(Pi) * n^(49/80)). - _Vaclav Kotesovec_, Nov 09 2017

%t nmax=29; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k^2 - 5 k + 2)/2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000335, A006564, A023872.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 18 2016