A396890 Convolution of A000290 and A000219.

0, 1, 5, 16, 43, 105, 239, 517, 1073, 2153, 4199, 7993, 14894, 27240, 48995, 86811, 151734, 261940, 447060, 755029, 1262778, 2092915, 3439512, 5607838, 9075151, 14583428, 23279896, 36929333, 58233142, 91307215, 142394398, 220923266, 341075091, 524095611, 801693228, 1221018870

Offset

0,3

Comments

In general, for m>=1, convolution of m-th powers and A000219 is asymptotic to Gamma(m+1) * exp(1/12 + 3*zeta(3)^(1/3)*n^(2/3)/2^(2/3)) * n^(m/3 - 13/36) / (A * sqrt(3*Pi) * 2^(23/36 + m/3) * zeta(3)^(5/36 + m/3)), where A is the Glaisher-Kinkelin constant (A074962).

Links

Table of n, a(n) for n=0..35.

Formula

G.f.: x*(1+x) / ((1-x)^3 * Product_{k>=1} (1-x^k)^k).

a(n) ~ exp(1/12 + 3*zeta(3)^(1/3)*n^(2/3)/2^(2/3)) * n^(11/36) / (A * 2^(11/36) * sqrt(3*Pi) * zeta(3)^(29/36)), where A is the Glaisher-Kinkelin constant (A074962).

Mathematica

nmax = 40; CoefficientList[Series[x*(1+x) / ((1-x)^3 * Product[(1-x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A000290, A000219, A074962, A360489.

Sequence in context: A321959 A066634 A241794 * A034358 A036888 A053221

Adjacent sequences: A396887 A396888 A396889 * A396891 A396892 A396893

Keyword

nonn

Author

Vaclav Kotesovec, Jun 09 2026

Status

approved