login
A360489
Convolution of A000219 and A001477.
0
0, 1, 3, 8, 19, 43, 91, 187, 369, 711, 1335, 2459, 4442, 7904, 13851, 23965, 40958, 69248, 115872, 192097, 315652, 514485, 832112, 1336214, 2131099, 3377178, 5319290, 8330147, 12973662, 20100411, 30986772, 47542096, 72609729, 110410791, 167186826, 252138816, 378781852
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
a(n) = Sum_{k=0..n} A000219(k) * (n-k).
G.f.: x/(1-x)^2 * Product_{k>=1} 1/(1 - x^k)^k.
a(n) ~ exp(1/12 + 3*zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * sqrt(3*Pi) * 2^(35/36) * zeta(3)^(17/36) * n^(1/36)), where A is the Glaisher-Kinkelin constant A074962.
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