A238209 - OEIS

%I #8 May 01 2020 17:33:41

%S 0,0,1,0,0,0,1,1,1,2,2,3,3,4,4,6,6,8,9,11,13,16,18,22,26,30,35,41,48,

%T 55,64,73,85,97,111,127,146,165,189,214,244,276,313,353,400,451,508,

%U 572,644,722,811,909,1018,1139,1273,1421,1586,1768,1968,2191,2436

%N The total number of 2'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="/A238209/b238209.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{j=1..round(n/4)} A067661(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067659(n-4*j).

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

%e a(11) = 3 because the partitions in question are: 8+2+1, 6+3+2, 5+4+2.

%o (PARI) seq(n)={my(A=O(x^(n-1))); Vec(x^2*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) + eta(x + A)/(1-x^2))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020

%Y Column k=2 of A238450.

%Y Cf. A067659, A067661.

%K nonn

%O 0,10

%A _Mircea Merca_, Feb 20 2014

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