A238215 - OEIS

%I #20 Nov 07 2024 06:41:59

%S 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 59,68,79,90,104,119,136,156,178,202,230,261,296,335,379,427,482,543,

%U 610,686,770,863,967,1082,1209,1351,1508,1681,1873,2085,2318,2577

%N The total number of 1'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="/A238215/b238215.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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 a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 07 2024

%e a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.

%p b:= proc(n, i, t) option remember; `if`(n=0, t,

%p      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))

%p     end:

%p a:= n-> b(n-1, 2, 0):

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, May 01 2020

%t 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 a[n_] := b[n - 1, 2, 0];

%t a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 01 2020, after _Alois P. Heinz_ *)

%o (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 Column k=1 of A238451.

%Y Cf. A067659, A067661.

%K nonn

%O 0,11

%A _Mircea Merca_, Feb 20 2014

%E Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020