cp's OEIS Frontend

A238219 The total number of 4's in all partitions of n into an even number of distinct parts.

%I A238219 #11 Jul 05 2025 08:54:50
%S A238219 0,0,0,0,0,1,1,1,0,1,2,1,2,3,4,4,5,6,8,9,11,13,16,18,21,25,29,34,40,
%T A238219 46,53,62,71,82,94,108,124,142,161,185,210,238,270,307,347,392,442,
%U A238219 499,562,632,709,797,894,1000,1119,1252,1398,1560,1739,1937,2157
%N A238219 The total number of 4's in all partitions of n into an even number of distinct parts.
%C A238219 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 A238219 Andrew Howroyd, <a href="/A238219/b238219.txt">Table of n, a(n) for n = 0..1000</a>
%F A238219 a(n) = Sum_{j=1..round(n/8)} A067659(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067661(n-8*j).
%F A238219 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 A238219 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238219 a(13) = 3 because the partitions in question are: 9+4, 6+4+2+1, 5+4+3+1.
%t A238219 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 A238219 Column k=4 of A238451.
%Y A238219 Cf. A067659, A067661.
%K A238219 nonn
%O A238219 0,11
%A A238219 _Mircea Merca_, Feb 20 2014
%E A238219 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020