A281156 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).

1, 1, 5, 19, 54, 165, 467, 1317, 3599, 9687, 25519, 66203, 169254, 426750, 1062950, 2616818, 6373911, 15369774, 36716706, 86939235, 204152395, 475631501, 1099874363, 2525418842, 5759549109, 13050991205, 29391523405, 65801951770, 146486952644, 324340095729, 714389015139

Offset: 0

Author

Ilya Gutkovskiy, Jan 16 2017

Keywords

• nonn

Comments

Weigh transform of square pyramidal numbers (A000330).

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, Square Pyramidal Number
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A000330, A027998, A258343.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k+1)/6).

a(n) ~ exp(5*(15*Zeta(5))^(1/5) * n^(4/5) / 2^(11/5) + 7*Pi^4 * n^(3/5) / (360*2^(2/5) * (15*Zeta(5))^(3/5)) + (Zeta(3) / (2^(13/5) * (15*Zeta(5))^(2/5)) - 49*Pi^8 / (2160000 * 2^(3/5) * 15^(2/5) * Zeta(5)^(7/5)))*n^(2/5) + (343*Pi^12 / (9720000000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + 49*Pi^8 * Zeta(3) / (129600000 * Zeta(5)^2) - 2401 * Pi^16 / (83980800000000 * Zeta(5)^3) - Zeta(3)^2 / (1200*Zeta(5))) * (3*Zeta(5))^(1/10) / (2^(11/18) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 09 2017