A301777 Expansion of Product_{k>=1} (1 + x^k)^A001001(k).

1, 1, 7, 20, 69, 178, 571, 1451, 4108, 10480, 27578, 68401, 172818, 417979, 1017575, 2410964, 5702481, 13228877, 30573978, 69594694, 157597162, 352694078, 784615466, 1728604925, 3785636280, 8221695626, 17751593170, 38051212654, 81103710142, 171757084527

Offset: 0

Author

Vaclav Kotesovec, Mar 26 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001001, A226313.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[Sum[d*DivisorSigma[1, d], {d, Divisors[k]}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

Formula

a(n) ~ exp(2*Pi^(3/2) * (7*Zeta(3))^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)) - 3*sqrt(5*Zeta(3)*n) / (4*7^(1/2)*Pi) + (sqrt(Pi) * 5^(1/4) / (3^(3/2) * (7*Zeta(3))^(1/4)) - 3^(5/2) * 5^(5/4) * Zeta(3)^(3/4) / (7^(5/4) * Pi^(7/2))) * n^(1/4) / 16 + 5/(448*Pi^2) - 675*Zeta(3) / (784*Pi^6)) * Pi^(1/4) * (7*Zeta(3))^(1/8) / (4*3^(1/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 26 2018