This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103258 #26 Jul 19 2025 13:07:04
%S A103258 1,2,4,8,12,20,32,48,72,106,152,216,304,420,576,784,1056,1412,1876,
%T A103258 2472,3240,4224,5472,7056,9056,11566,14712,18640,23520,29572,37056,
%U A103258 46272,57600,71488,88456,109152,134332,164884,201888,246608,300528,365428,443392,536856
%N A103258 G.f. = theta_4(0,x^4)/theta_4(0,x).
%C A103258 G.f. for the number of partitions of 2n in which all odd parts occur with multiplicities 2, 4 or 6. The even parts appear at most three times. E.g., a(8)=12 because 8 = 6+2 = 6+1+1 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1.
%C A103258 Also the number of partitions of 2n in which the even parts appear with 2 types c, c* and with multiplicity 1. The odd parts with multiplicity 4. E.g., a(6)=8 because we have 6, 6*, 42, 42*, 4*2, 4*2*, 21111, 2*1111.
%H A103258 Vaclav Kotesovec, <a href="/A103258/b103258.txt">Table of n, a(n) for n = 0..1000</a>
%H A103258 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See pp. 4-5, 9, 14.
%H A103258 Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%F A103258 Expansion of eta(q^2)*eta(q^4)^2/(eta(q)^2*eta(q^8)) in powers of q.
%F A103258 Euler transform of period 8 sequence [2, 1, 2, -1, 2, 1, 2, 0, ...]. - _Michael Somos_, Feb 10 2005
%F A103258 G.f.: product_{k>0} ((1+x^k)^2 * (1+x^(2*(2*k-1)))).
%F A103258 From _Vaclav Kotesovec_, Jan 10 2017: (Start)
%F A103258 a(n) ~ sqrt(3)*Pi * BesselI(1, sqrt(3*n)*Pi/2) / (8*sqrt(n)).
%F A103258 a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n)/2) / (8*n^(3/4)) * (1 - sqrt(3)/(4*Pi*sqrt(n)) - 5/(32*Pi^2*n)).
%F A103258 (End)
%t A103258 nmax = 60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(8*k-4)) * (1+x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%o A103258 (PARI) {a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^4+A)^2/eta(x+A)^2/eta(x^8+A), n))} /* _Michael Somos_, Feb 10 2005 */
%Y A103258 Cf. A002448.
%K A103258 nonn
%O A103258 0,2
%A A103258 _Noureddine Chair_, Jan 27 2005