A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k^2+1)/3).

1, 1, 7, 26, 91, 290, 946, 2922, 8937, 26521, 77485, 222005, 626988, 1743739, 4787625, 12979799, 34792728, 92257673, 242197348, 629805075, 1623197726, 4148192991, 10516418844, 26458470616, 66086152465, 163925621199, 403931474096, 989040788801, 2407020523315, 5823830868091, 14011949899801, 33530477120905, 79820957945103

Offset: 0

Author

Ilya Gutkovskiy, Dec 18 2016

Keywords

• nonn

Comments

Euler transform of the octahedral numbers (A005900).

Links

• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Octahedral Number
• OEIS Wiki, Platonic numbers

Crossrefs

Cf. A000335, A005900, A023872.

Programs

Mathematica

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(2*k^2+1)/3).

a(n) ~ exp(Zeta'(-1)/3 - Zeta(3)^2 / (360*Zeta(5)) + 2*Zeta'(-3)/3 + (Zeta(3)/(6*2^(3/5) * Zeta(5)^(2/5))) * n^(2/5) + (5*(Zeta(5)/2)^(1/5)/2) * n^(4/5)) * Zeta(5)^(47/450) / (2^(37/450) * sqrt(5*Pi) * n^(136/225)). - Vaclav Kotesovec, Nov 09 2017