A360489 Convolution of A000219 and A001477.

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

Author

Vaclav Kotesovec, Feb 09 2023

Keywords

• nonn

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).

Crossrefs

Cf. A000219, A001477, A014153, A095944, A161870.

Programs

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]

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.