A333179 G.f.: Sum_{k>=0} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 5, 5, 4, 4, 4, 4, 5, 6, 5, 6, 7, 7, 8, 8, 8, 8, 9, 8, 8, 8, 7, 8, 8, 8, 8, 9, 9, 10, 11, 11

Offset: 0

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A003106, A053261, A306734.

Programs

Mathematica

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

Formula

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.3207396095989103757477946185... = sqrt((1 - (2/(23*(23 + 3*sqrt(69))))^(1/3) + ((1/2)*(23 + 3*sqrt(69)))^(1/3)/23^(2/3))/3)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 8*s - 23*s^2 + 23*s^3 = 0.

Limit_{n->infinity} A306734(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...