OFFSET
0,3
COMMENTS
Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (1/n)*Sum_{k=1..n} sigma(k^2) * a(n-k) for n>0, with a(0)=1.
Euler transform of Dedekind psi function, cf. A001615. - Vladeta Jovovic, Feb 12 2009
a(n) ~ exp(3^(4/3) * (5*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) - Pi^(2/3) * n^(1/3) / (2^(4/3) * (15*Zeta(3))^(1/3)) + 1/12 - Pi^2 / (720 * Zeta(3))) * (5*Zeta(3))^(7/36) / (A * 2^(2/9) * 3^(11/36) * Pi^(29/36) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 24 2018
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 20*x^4 + 38*x^5 + 88*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 31*x^4/4 + 31*x^5/5 + 127*x^6/6 +...
MATHEMATICA
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*Sum[MoebiusMu[d]^2 / 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 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^2)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^2)*a(n-k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2009
STATUS
approved