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 A238210 #14 Jul 05 2025 08:44:50
%S A238210 0,0,0,1,0,0,1,0,1,2,2,2,3,3,4,5,6,7,8,10,12,14,17,20,23,28,32,37,44,
%T A238210 51,58,68,78,89,103,118,134,154,175,199,227,257,291,330,373,421,475,
%U A238210 535,602,677,760,852,955,1069,1196,1336,1491,1663,1853,2063,2296
%N A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.
%C A238210 The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).
%H A238210 Andrew Howroyd, <a href="/A238210/b238210.txt">Table of n, a(n) for n = 0..1000</a>
%F A238210 a(n) = Sum_{j=1..round(n/6)} A067661(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067659(n-6*j).
%F A238210 G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
%F A238210 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238210 a(12) = 3 because the partitions in question are: 8+3+1, 7+3+2, 5+4+3.
%t A238210 nmax = 100; With[{k=3}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%o A238210 (PARI) seq(n)={my(A=O(x^(n-2))); Vec(x^3*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) + eta(x + A)/(1-x^3))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020
%Y A238210 Column k=3 of A238450.
%Y A238210 Cf. A067659, A067661.
%K A238210 nonn
%O A238210 0,10
%A A238210 _Mircea Merca_, Feb 20 2014
%E A238210 Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020