login
Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^3))).
5

%I #17 Mar 03 2024 11:16:27

%S 1,2,4,7,12,19,30,45,68,99,143,202,284,392,538,729,983,1311,1740,2289,

%T 2998,3898,5046,6492,8321,10607,13472,17032,21460,26927,33682,41975,

%U 52160,64600,79790,98255,120690,147836,180662,220217,267841,324999,393539,475496,573403

%N Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^3))).

%C Convolution of A000041 and A003108.

%C a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k and P(n-k) is a partition of n-k into cubes.

%H Vaclav Kotesovec, <a href="/A369579/b369579.txt">Table of n, a(n) for n = 0..10000</a>

%H Vaclav Kotesovec, <a href="/A369579/a369579.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%F a(n) ~ exp(Pi*sqrt(2*n/3) + 6^(1/6) * Gamma(4/3) * zeta(4/3) * n^(1/6) / Pi^(1/3)) / (2^(15/4) * 3^(3/4) * Pi * n^(5/4)) * (1 - Gamma(1/3)^2 * zeta(4/3)^2 / (54 * 6^(1/6) * Pi^(5/3) * n^(1/6))).

%t nmax = 50; CoefficientList[Series[Product[1/((1-x^k)*(1-x^(k^3))), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000041, A001156, A003108, A280278, A369519, A369520.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 26 2024