A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).
1, 1, 7, 27, 98, 323, 1085, 3471, 10998, 33874, 102737, 305849, 897899, 2597822, 7423408, 20957775, 58524868, 161741013, 442705279, 1200718351, 3228796864, 8611973548, 22793714865, 59887897679, 156252738062, 404964879419, 1042884107691, 2669317020743, 6792321636929
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..2000
- N. J. A. Sloane, Transforms
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d^2* (d+2)*(d+1)^2/12, d=numtheory[divisors](j)), j=1..n)/n) end: seq(a(n), n=0..35); # Alois P. Heinz, Jun 07 2018 -
Mathematica
nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((k + 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2 (d + 2)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^A002415(k+1).
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^5)).
a(n) ~ exp(Zeta'(-1)/6 - Zeta(3) / (4*Pi^2) + 149*Zeta(5) / (32*Pi^4) + 15876 * Zeta(3) * Zeta(5)^2 / Pi^12 - 666792 * Zeta(5)^3 / Pi^14 + 108884466432 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/3 + (-7*(7/2)^(1/6) * Pi / (384*sqrt(3)) - 21 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / Pi^7 + 3087 * sqrt(3) * (7/2)^(1/6) * Zeta(5)^2 / (2*Pi^9) - 30339036 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(5)^4 / Pi^19) * n^(1/6) + ((7/2)^(1/3) * Zeta(3) / (2*Pi^2) - 21 * (7/2)^(1/3) * Zeta(5) / (2*Pi^4) + 254016 * 2^(2/3) * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (sqrt(7/6) * Pi / 12 - 756 * sqrt(42) * Zeta(5)^2 / Pi^9) * sqrt(n) + (9 * 2^(1/3) * 7^(2/3) * Zeta(5) / Pi^4) * n^(2/3) + (2 * (2/7)^(1/6) * sqrt(3) * Pi) / 5 * n^(5/6)) * Pi^(1/90) / (2^(247/270) * 3^(34/45) * 7^(23/270) * n^(79/135)). - Vaclav Kotesovec, Jun 08 2018
Comments