OFFSET
0,3
COMMENTS
In general, for 0 < p < 1, delta > 1, beta > -1, the convolution of (delta^(n^p) * n^alfa) and n^beta is asymptotic to delta^(n^p) * n^(alfa + (1-p)*(beta+1)) * Gamma(beta+1) / (p^(beta+1) * log(delta)^(beta+1)).
For p = 1 is the convolution of (delta^(n^p) * n^alfa) and n^beta asymptotic to delta^n * n^alfa * polylog(-beta, 1/delta).
FORMULA
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
end:
a:= n-> add(b(n-j)*j, j=0..n):
seq(a(n), n=0..42); # Alois P. Heinz, Feb 09 2023
MATHEMATICA
nmax = 50; CoefficientList[Series[x/(1-x)^2 * Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Feb 09 2023
STATUS
approved