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 A238212 #14 Jul 05 2025 08:45:25
%S A238212 0,0,0,0,0,1,0,0,1,1,2,1,2,2,3,5,4,5,7,8,10,11,13,16,19,23,26,31,36,
%T A238212 42,49,56,65,75,86,100,114,130,149,170,193,220,250,283,321,363,410,
%U A238212 463,522,587,660,742,832,933,1045,1168,1307,1459,1627,1814,2020
%N A238212 The total number of 5's in all partitions of n into an odd number of distinct parts.
%C A238212 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 A238212 Andrew Howroyd, <a href="/A238212/b238212.txt">Table of n, a(n) for n = 0..1000</a>
%F A238212 a(n) = Sum_{j=1..round(n/10)} A067661(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067659(n-10*j).
%F A238212 G.f.: (1/2)*(x^5/(1+x^5))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^5/(1-x^5))*(Product_{n>=1} 1 - x^n).
%F A238212 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238212 a(12) = 2 because the partitions in question are: 6+5+1, 5+4+3.
%t A238212 tn5[n_]:=Module[{op=IntegerPartitions[n],m},m=Flatten[Select[op,OddQ[ Length[#]] && Length[#]==Length[Union[#]]&]];Count[m,5]]; Array[tn5,60,0] (* _Harvey P. Dale_, Feb 06 2015 *)
%t A238212 nmax = 100; With[{k=5}, 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 A238212 Column k=5 of A238450.
%Y A238212 Cf. A067659, A067661.
%K A238212 nonn
%O A238212 0,11
%A A238212 _Mircea Merca_, Feb 20 2014
%E A238212 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020