cp's OEIS Frontend

A238220 The total number of 5's in all partitions of n into an even number of distinct parts.

%I A238220 #14 Jul 05 2025 08:54:54
%S A238220 0,0,0,0,0,0,1,1,1,1,0,2,2,2,3,3,5,6,7,8,9,12,14,16,19,22,27,31,36,42,
%T A238220 48,56,65,75,86,99,114,130,149,170,193,220,250,283,321,364,410,463,
%U A238220 522,587,661,742,832,933,1045,1169,1306,1459,1627,1814,2021
%N A238220 The total number of 5's in all partitions of n into an even number of distinct parts.
%C A238220 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 A238220 Andrew Howroyd, <a href="/A238220/b238220.txt">Table of n, a(n) for n = 0..1000</a>
%F A238220 a(n) = Sum_{j=1..round(n/10)} A067659(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067661(n-10*j).
%F A238220 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 A238220 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238220 a(13) = 2 because the partitions in question are: 8+5, 5+4+3+1.
%t A238220 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 A238220 Column k=5 of A238451.
%Y A238220 Cf. A067659, A067661.
%K A238220 nonn
%O A238220 0,12
%A A238220 _Mircea Merca_, Feb 20 2014
%E A238220 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020