A306734 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j).

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 8, 9, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 14, 15, 15, 15, 15, 15, 15, 16

Offset: 0

Author

Ilya Gutkovskiy, Mar 06 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A003114, A053256, A053258, A053261, A333179, A333180.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Sum[x^(k^2) Product[(1 + x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 10 2020 *)

Formula

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.424889520435345887204307524... = sqrt((23 + (10051/2 - (1173*sqrt(69))/2)^(1/3) + ((23/2)*(437 + 51*sqrt(69)))^(1/3))/69)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 6*s - 23*s^2 + 23*s^3 = 0. - Vaclav Kotesovec, Mar 11 2020

Limit_{n->infinity} a(n) / A333179(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885... - Vaclav Kotesovec, Mar 11 2020