A195734 G.f.: exp( Sum_{n>=1} (2*sigma(n^2) - sigma(n)^2) * x^n/n ).
1, 1, 3, 6, 11, 22, 40, 72, 123, 215, 363, 605, 991, 1618, 2598, 4139, 6537, 10229, 15871, 24476, 37487, 56995, 86177, 129531, 193662, 287992, 426254, 627841, 920708, 1344331, 1954987, 2831688, 4086168, 5875087, 8417724, 12020250, 17108958, 24275947, 34340966
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 22*x^5 + 40*x^6 + 72*x^7 +... where log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 13*x^4/4 + 26*x^5/5 + 38*x^6/6 + 50*x^7/7 + 29*x^8/8 +...+ A195735(n)*x^n/n +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)
Programs
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Mathematica
nmax = 40; CoefficientList[Series[Exp[Sum[(2*DivisorSigma[1,k^2] - DivisorSigma[1,k]^2) * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2024 *) -
PARI
{a(n)=polcoeff(exp(sum(k=1, n,(2*sigma(k^2)-sigma(k)^2)*x^k/k)+x*O(x^n)), n)}
Formula
Logarithmic derivative equals A195735.
log(a(n)) ~ 3*(5*zeta(3)*(12 - Pi^2))^(1/3) * n^(2/3) / (2*Pi^(2/3)). - Vaclav Kotesovec, Oct 31 2024