OFFSET
0,3
COMMENTS
In general, if m >= 1 and g.f.= exp(Sum_{k>=1} sigma(k^m)*x^k/k), then log(a(n)) ~ (1 + 1/m) * (c*m!)^(1/(m+1)) * n^(m/(m+1)), where c = Product_{primes p} ((p^(m+2) - p^(m+1) + p^m - p) / ((p-1)*(p^(m+1)-1))). - Vaclav Kotesovec, Nov 01 2024
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
Logarithmic derivative yields A203556.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A203556(k)*a(n-k) for n > 0. - Seiichi Manyama, Sep 09 2020
log(a(n)) ~ 2^(3/2) * 3^(7/6) * c^(1/6) * n^(5/6) / 5^(5/6), where c = Product_{primes p} (p*(1 + p + p^2 + p^3 + p^5) / (p^6 - 1)) = 1.93252811194652723494722635658171746713... - Vaclav Kotesovec, Nov 01 2024
EXAMPLE
G.f.: A(x) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
where the logarithm equals the l.g.f. of A203556:
log(A(x)) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...+ sigma(n^5)*x^n/n +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^5)*x^m/m)+x*O(x^n)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2012
STATUS
approved