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 A238211 #14 Jul 05 2025 08:45:08
%S A238211 0,0,0,0,1,0,0,1,1,1,1,2,3,3,3,5,5,6,7,9,11,13,15,18,21,25,29,34,40,
%T A238211 46,54,62,71,82,95,108,124,142,162,184,210,238,271,306,346,392,443,
%U A238211 498,561,632,710,796,893,1000,1120,1252,1397,1560,1740,1937,2156
%N A238211 The total number of 4's in all partitions of n into an odd number of distinct parts.
%C A238211 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 A238211 Andrew Howroyd, <a href="/A238211/b238211.txt">Table of n, a(n) for n = 0..1000</a>
%F A238211 a(n) = Sum_{j=1..round(n/8)} A067661(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067659(n-8*j).
%F A238211 G.f.: (1/2)*(x^4/(1+x^4))*(Product{n>=1} 1 + x^n) + (1/2)*(x^4/(1-x^4))*(Product_{n>=1} 1 - x^n).
%F A238211 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238211 a(12) = 3 because the partitions in question are: 7+4+1, 6+4+2, 5+4+3.
%t A238211 nmax = 100; With[{k=4}, 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 *)
%Y A238211 Column k=4 of A238450.
%Y A238211 Cf. A067659, A067661.
%K A238211 nonn
%O A238211 0,12
%A A238211 _Mircea Merca_, Feb 20 2014
%E A238211 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020