A238221 - OEIS

%I #11 Apr 29 2020 18:18:24

%S 0,0,0,0,0,0,0,1,1,1,1,1,1,2,3,3,4,4,6,7,9,11,12,14,17,20,24,28,32,37,

%T 44,51,59,69,78,90,104,119,136,156,177,202,230,261,296,336,379,428,

%U 483,544,612,689,773,867,972,1088,1217,1360,1518,1693,1887

%N The total number of 6'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="/A238221/b238221.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{j=1..round(n/12)} A067659(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067661(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(13) = 2 because the partitions in question are: 7+6, 6+4+2+1.

%t endpQ[n_]:=Module[{len=Length[n]},EvenQ[len]&&len==Length[Union[n]]]; Table[ Count[Flatten[Select[IntegerPartitions[i],endpQ]],6],{i,0,50}] (* _Harvey P. Dale_, Mar 03 2014 *)

%Y Column k=6 of A238451.

%Y Cf. A067659, A067661.

%K nonn

%O 0,14

%A _Mircea Merca_, Feb 20 2014

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