A301746 Expansion of Product_{k>=1} (1 + x^k)^(sigma_0(k)^2).

1, 1, 4, 8, 19, 35, 82, 142, 291, 524, 989, 1724, 3174, 5393, 9517, 16064, 27464, 45481, 76357, 124402, 204497, 329559, 532316, 846564, 1349481, 2120814, 3335819, 5191522, 8070062, 12434176, 19136484, 29215324, 44531151, 67431985, 101882975, 153055897

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

Conjecture: log(a(n)) ~ sqrt(n) * log(n)^(3/2) / (2*sqrt(6)). - Vaclav Kotesovec, Aug 29 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^k)^(DivisorSigma[0, k]^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]^2, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 29 2018 *)

Crossrefs

Cf. A000005, A035116, A107742, A301747.

Sequence in context: A274817 A130887 A049933 * A163318 A129362 A301981

Adjacent sequences: A301743 A301744 A301745 * A301747 A301748 A301749

Keyword

nonn

Author

Vaclav Kotesovec, Mar 26 2018

Status

approved