cp's OEIS Frontend

A238221 The total number of 6's in all partitions of n into an even number of distinct parts.

%I A238221 #14 Jul 05 2025 08:54:57
%S A238221 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 A238221 44,51,59,69,78,90,104,119,136,156,177,202,230,261,296,336,379,428,
%U A238221 483,544,612,689,773,867,972,1088,1217,1360,1518,1693,1887
%N A238221 The total number of 6's in all partitions of n into an even number of distinct parts.
%C A238221 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 A238221 Andrew Howroyd, <a href="/A238221/b238221.txt">Table of n, a(n) for n = 0..1000</a>
%F A238221 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 A238221 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).
%F A238221 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238221 a(13) = 2 because the partitions in question are: 7+6, 6+4+2+1.
%t A238221 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 *)
%t A238221 nmax = 100; With[{k=6}, 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 A238221 Column k=6 of A238451.
%Y A238221 Cf. A067659, A067661.
%K A238221 nonn
%O A238221 0,14
%A A238221 _Mircea Merca_, Feb 20 2014
%E A238221 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020