A281157 Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k^2+1)/3).

1, 1, 6, 25, 78, 258, 800, 2463, 7344, 21511, 61677, 173980, 483319, 1323470, 3577605, 9553658, 25227727, 65918419, 170552866, 437196640, 1110945961, 2799689792, 7000246591, 17372882671, 42809388080, 104774554942, 254771953179, 615667051237, 1478934870484, 3532347875968

Offset: 0

Author

Ilya Gutkovskiy, Jan 16 2017

Keywords

• nonn

Comments

Weigh transform of 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

Crossrefs

Cf. A005900, A248882, A258343.

Programs

Mathematica

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ exp(-Zeta(3)^2 / (600*Zeta(5)) + (Zeta(3) / (4*(15*Zeta(5))^(2/5))) * n^(2/5) + (5*(15*Zeta(5))^(1/5) / 4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (sqrt(Pi) * 2^(47/90) * 5^(2/5) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017