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A238215 The total number of 1's in all partitions of n into an even number of distinct parts.

%I A238215 #20 Nov 07 2024 06:41:59
%S A238215 0,0,0,1,1,1,1,1,1,1,2,2,3,4,5,6,8,9,11,13,15,18,21,24,28,33,38,44,51,
%T A238215 59,68,79,90,104,119,136,156,178,202,230,261,296,335,379,427,482,543,
%U A238215 610,686,770,863,967,1082,1209,1351,1508,1681,1873,2085,2318,2577
%N A238215 The total number of 1's in all partitions of n into an even number of distinct parts.
%C A238215 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 A238215 Andrew Howroyd, <a href="/A238215/b238215.txt">Table of n, a(n) for n = 0..1000</a>
%F A238215 a(n) = Sum_{j=1..round(n/2)} A067659(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067661(n-2*j).
%F A238215 G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) - (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).
%F A238215 a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2024
%e A238215 a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.
%p A238215 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A238215      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
%p A238215     end:
%p A238215 a:= n-> b(n-1, 2, 0):
%p A238215 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 01 2020
%t A238215 b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 1, 1 - t]]];
%t A238215 a[n_] := b[n - 1, 2, 0];
%t A238215 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 01 2020, after _Alois P. Heinz_ *)
%o A238215 (PARI) seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) - eta(x + A)/(1-x))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020
%Y A238215 Column k=1 of A238451.
%Y A238215 Cf. A067659, A067661.
%K A238215 nonn
%O A238215 0,11
%A A238215 _Mircea Merca_, Feb 20 2014
%E A238215 Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020