A294845 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k-1)/2).

1, 1, 9, 39, 136, 511, 1785, 6139, 20404, 66406, 211418, 660752, 2030172, 6139231, 18300573, 53823451, 156344596, 448886205, 1274840165, 3583595734, 9976530997, 27520998775, 75262394273, 204130567402, 549318633095, 1467178746342, 3890697051314, 10246833932820, 26809705578787, 69702402930045

Offset: 0

Author

Ilya Gutkovskiy, Nov 09 2017

Keywords

• nonn

Comments

Weigh transform of the octagonal pyramidal numbers (A002414).

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

Crossrefs

Cf. A002414, A258343, A281156, A294838, A294841, A294842, A294843, A294844.

Programs

Mathematica

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1) (2 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A002414(k).

a(n) ~ exp(-2401 * Pi^16 / (2267481600000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (388800000 * Zeta(5)^2) - Zeta(3)^2 / (400*Zeta(5)) + (343*Pi^12 / (87480000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (6480000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5)*Zeta(3) / (2^(13/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (1080 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(11/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(11/20) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017