A333180 G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 6, 3, 3, 3, 4, 4, 4, 8, 8, 8, 8, 8, 4, 9, 9, 5, 10, 10, 15, 15, 15, 15, 15, 15, 16, 16, 11, 17, 17, 18, 24, 24, 24, 30, 30, 30, 30, 31, 31, 31, 32, 26, 33, 34, 41, 41, 42, 49, 49, 56, 56, 56, 64, 64, 57, 65, 58, 59, 67, 68

Offset: 0

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A268188, A306734, A333181.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[n*x^(n^2)*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += k*p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ c * A333198^sqrt(n), where c = 0.3836313809149103736315...

Limit_{n->infinity} a(n) / A333181(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...