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The total number of 4's in all partitions of n into an odd number of distinct parts.
2

%I #11 Apr 29 2020 18:16:33

%S 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 46,54,62,71,82,95,108,124,142,162,184,210,238,271,306,346,392,443,

%U 498,561,632,710,796,893,1000,1120,1252,1397,1560,1740,1937,2156

%N The total number of 4's in all partitions of n into an odd 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="/A238211/b238211.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 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(12) = 3 because the partitions in question are: 7+4+1, 6+4+2, 5+4+3.

%Y Column k=4 of A238450.

%Y Cf. A067659, A067661.

%K nonn

%O 0,12

%A _Mircea Merca_, Feb 20 2014

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