A240011 - OEIS

%I #15 Dec 10 2020 17:37:00

%S 1,0,1,1,2,2,3,4,5,8,8,13,14,21,23,34,37,52,60,79,93,120,143,178,216,

%T 263,321,386,470,560,684,806,980,1154,1395,1636,1969,2304,2758,3225,

%U 3835,4480,5305,6186,7288,8495,9961,11594,13545,15742,18325,21269,24675

%N Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 2.

%C With offset 4 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -2.

%H Alois P. Heinz, <a href="/A240011/b240011.txt">Table of n, a(n) for n = 2..1000</a>

%e a(10) = 5: [9,1], [7,3], [5,5], [4,2,1,1,1,1], [3,2,2,1,1,1].

%e a(11) = 8: [8,1,1,1], [7,2,1,1], [6,3,1,1], [5,4,1,1], [5,3,2,1], [4,3,3,1], [3,3,3,2], [2,2,2,1,1,1,1,1].

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

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

%p       `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))

%p     end:

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

%p seq(a(n), n=2..80);

%t b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];

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

%t a /@ Range[2, 80] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A240009.

%K nonn

%O 2,5

%A _Alois P. Heinz_, Mar 30 2014