A333181 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 3, 3, 3, 6, 3, 3, 3, 0, 4, 4, 4, 8, 8, 8, 8, 8, 4, 4, 9, 5, 5, 10, 10, 15, 15, 15, 15, 15, 15, 10, 16, 11, 11, 17, 12, 18, 24, 24, 24, 30, 30, 30, 30, 24, 31, 31, 25, 26, 26, 27, 34, 41, 35, 42, 49, 49, 56, 56, 56, 56, 64

Offset: 0

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A333153, A333179, A333180.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[n*x^(n*(n+1))*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)]; 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 = 0.2895947615240435716456...

Limit_{n->infinity} A333180(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...