A294843 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(4*k-1)/6).

1, 1, 7, 29, 93, 320, 1026, 3256, 9995, 30102, 88722, 257042, 732876, 2058370, 5703858, 15606076, 42203027, 112882223, 298849221, 783574536, 2035876825, 5244191462, 13398463986, 33967008194, 85476285603, 213583335753, 530099612487, 1307195997381, 3203555001240, 7804386224233

Offset

0,3

Comments

Weigh transform of the hexagonal pyramidal numbers (A002412).

Links

Table of n, a(n) for n=0..29.

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, Hexagonal Pyramidal Number

Formula

G.f.: Product_{k>=1} (1 + x^k)^A002412(k).

a(n) ~ exp(-2401 * Pi^16 / (671846400000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (518400000 * Zeta(5)^2) - Zeta(3)^2 / (2400*Zeta(5)) + (343 * Pi^12 / (77760000000 * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4*Zeta(3) / (72000 * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (8640000 * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (8 * (15*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (720 * (15*Zeta(5))^(3/5))) * n^(3/5) + (5*(15*Zeta(5))^(1/5)/4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (2^(173/360) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017

Mathematica

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1)(4 k - 1)/6), {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)(4 d - 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

Crossrefs

Cf. A002412, A028377, A258343, A281156, A294836, A294842, A294844, A294845.

Sequence in context: A055798 A002664 A290901 * A042609 A002941 A193655

Adjacent sequences: A294840 A294841 A294842 * A294844 A294845 A294846

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 09 2017

Status

approved