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A275585
Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).
14
1, 1, 6, 16, 52, 128, 373, 913, 2399, 5796, 14298, 33655, 79756, 183078, 419846, 942807, 2106176, 4633208, 10127557, 21870997, 46912648, 99639685, 210206722, 439777198, 914157490, 1886428608, 3869204040, 7884691072, 15976273573, 32182538964, 64484592372, 128518359868, 254868985099, 502950483815, 987904826874, 1931596634076
OFFSET
0,3
COMMENTS
Euler transform of the sum of squares of divisors (A001157).
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
FORMULA
G.f.: Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 08 2017
a(n) ~ exp(4*Pi * Zeta(3)^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi * 5^(1/4) * n^(1/4) / (8 * 3^(7/4) * Zeta(3)^(1/4)) + Zeta(3) / (8*Pi^2)) * Zeta(3)^(1/8) / (2^(3/2) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*sigma[2](d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jun 08 2017
MATHEMATICA
nmax = 35; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[2, k]), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), this sequence (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).
Sequence in context: A288572 A367993 A301978 * A026086 A175659 A221270
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 25 2016
STATUS
approved