A238219 - OEIS

%I #8 Apr 29 2020 18:17:35

%S 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 46,53,62,71,82,94,108,124,142,161,185,210,238,270,307,347,392,442,

%U 499,562,632,709,797,894,1000,1119,1252,1398,1560,1739,1937,2157

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

%C 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 Andrew Howroyd, <a href="/A238219/b238219.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 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).

%e a(13) = 3 because the partitions in question are: 9+4, 6+4+2+1, 5+4+3+1.

%Y Column k=4 of A238451.

%Y Cf. A067659, A067661.

%K nonn

%O 0,11

%A _Mircea Merca_, Feb 20 2014

%E Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020