A240140 - OEIS

%I #11 Dec 10 2020 17:34:38

%S 1,0,1,0,2,0,3,0,5,0,6,1,9,2,11,4,15,7,18,12,23,19,27,29,34,42,40,60,

%T 49,83,59,113,73,150,89,197,112,254,141,324,180,408,231,509,298,629,

%U 386,771,500,938,648,1135,835,1365,1076,1634,1376,1949,1755,2317

%N Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 4.

%C With offset 20 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -4.

%H Alois P. Heinz, <a href="/A240140/b240140.txt">Table of n, a(n) for n = 16..1000</a>

%F a(n) = [x^n y^4] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

%e a(28) = 9: [19,5,3,1], [17,7,3,1], [15,9,3,1], [15,7,5,1], [13,11,3,1], [13,9,5,1], [13,7,5,3], [11,9,7,1], [11,9,5,3].

%e a(29) = 2: [11,7,5,3,2,1], [9,7,5,4,3,1].

%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or

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

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

%p     end:

%p a:= n-> b(n$2, -4):

%p seq(a(n), n=16..80);

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

%t a[n_] := b[n, n, -4];

%t a /@ Range[16, 80] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)

%Y Column k=4 of A240021.

%K nonn

%O 16,5

%A _Alois P. Heinz_, Apr 02 2014