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 A238209 #11 Jul 05 2025 08:44:33
%S A238209 0,0,1,0,0,0,1,1,1,2,2,3,3,4,4,6,6,8,9,11,13,16,18,22,26,30,35,41,48,
%T A238209 55,64,73,85,97,111,127,146,165,189,214,244,276,313,353,400,451,508,
%U A238209 572,644,722,811,909,1018,1139,1273,1421,1586,1768,1968,2191,2436
%N A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.
%C A238209 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 A238209 Andrew Howroyd, <a href="/A238209/b238209.txt">Table of n, a(n) for n = 0..1000</a>
%F A238209 a(n) = Sum_{j=1..round(n/4)} A067661(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067659(n-4*j).
%F A238209 G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
%F A238209 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238209 a(11) = 3 because the partitions in question are: 8+2+1, 6+3+2, 5+4+2.
%t A238209 nmax = 100; With[{k=2}, 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 A238209 (PARI) seq(n)={my(A=O(x^(n-1))); Vec(x^2*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) + eta(x + A)/(1-x^2))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020
%Y A238209 Column k=2 of A238450.
%Y A238209 Cf. A067659, A067661.
%K A238209 nonn
%O A238209 0,10
%A A238209 _Mircea Merca_, Feb 20 2014
%E A238209 Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020