A268499 - OEIS

%I #10 Apr 18 2026 17:33:27

%S 1,2,0,8,-2,8,16,8,8,10,80,-8,72,-24,144,128,134,40,224,120,232,688,

%T 176,696,32,1194,-96,1840,1144,2248,288,2968,800,4160,752,5104,6438,

%U 4984,5104,5488,10960,4856,14080,3480,24408,15448,26832,7080,42120,11178

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

%C In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 + x^k)) then a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt((m+1)*Pi) * n^(3/4)),  where c = Pi^2/3 + 2*log(m)^2 + 4*polylog(2, -1/m).

%H Vaclav Kotesovec, <a href="/A268499/b268499.txt">Table of n, a(n) for n = 0..5000</a>

%H Subhash Chand Bhoria, Pramod Eyyunni, and Subhrangsu Santra, <a href="https://arxiv.org/abs/2604.12557">On the number of missing integers in partitions</a>, arXiv:2604.12557 [math.CO], 2026. See pp. 13, 17.

%F a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(3)^2 + 4*polylog(2, -1/3) = 4.467633549370382939364... .

%t nmax = 100; CoefficientList[Series[Product[(1+3*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A032308, A266821, A268498, A268500.

%K sign

%O 0,2

%A _Vaclav Kotesovec_, Feb 06 2016