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The total number of 6's in all partitions of n into an odd number of distinct parts.
2

%I #9 Apr 29 2020 18:17:16

%S 0,0,0,0,0,0,1,0,0,1,1,2,2,2,2,3,4,5,6,7,8,10,12,15,17,20,23,27,33,38,

%T 44,51,59,68,79,91,104,119,136,155,178,202,230,261,296,335,379,428,

%U 483,544,612,688,773,867,972,1088,1217,1360,1518,1693,1887

%N The total number of 6'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="/A238213/b238213.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{j=1..round(n/12)} A067661(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067659(n-12*j).

%F G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).

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

%Y Column k=6 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