A294836 Expansion of Product_{k>=1} (1 + x^k)^(k*(2*k-1)).

1, 1, 6, 21, 58, 178, 494, 1365, 3640, 9533, 24401, 61384, 151958, 370335, 890565, 2113913, 4959199, 11505799, 26420628, 60082005, 135386341, 302448477, 670148898, 1473387787, 3215519032, 6968266907, 14999453058, 32079714584, 68187859040, 144083404856, 302727633735, 632579826174

Offset

0,3

Comments

Weigh transform of the hexagonal numbers (A000384).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(2*n-1), g(n) = -1. - Seiichi Manyama, Nov 14 2017

Links

Seiichi Manyama, Table of n, a(n) for n = 0..9392

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 Number

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000384(k).

a(n) ~ 7^(1/8) * exp(Pi*2^(3/2) * (7/15)^(1/4) * n^(3/4)/3 - 3*Zeta(3)*sqrt(15*n/7) / (2*Pi^2) - 135*Zeta(3)^2 * (15*n/7)^(1/4) / (28*sqrt(2)*Pi^5) - 2025*Zeta(3)^3 / (196*Pi^8)) / (2^(5/3) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(2*d-1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

Mathematica

nmax = 31; CoefficientList[Series[Product[(1 + x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

Crossrefs

Cf. A000384, A027998, A028377, A294102, A294837, A294838.

Sequence in context: A053809 A290891 A047520 * A341221 A143115 A258142

Adjacent sequences: A294833 A294834 A294835 * A294837 A294838 A294839

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 09 2017

Status

approved