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A240139 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 3.

%I A240139 #11 Dec 10 2020 17:34:54
%S A240139 1,0,1,0,2,0,3,0,4,1,5,2,7,4,8,7,10,12,12,18,14,27,17,38,21,53,26,71,
%T A240139 33,94,44,121,58,155,79,194,107,241,146,296,197,361,267,436,355,525,
%U A240139 472,628,618,750,805,894,1035,1064,1324,1267,1673,1511,2103,1804
%N A240139 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 3.
%C A240139 With offset 12 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -3.
%H A240139 Alois P. Heinz, <a href="/A240139/b240139.txt">Table of n, a(n) for n = 9..1000</a>
%F A240139 a(n) = [x^n y^3] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).
%e A240139 a(20) = 2: [9,5,3,2,1], [7,5,4,3,1].
%e A240139 a(21) = 7: [17,3,1], [15,5,1], [13,7,1], [13,5,3], [11,9,1], [11,7,3], [9,7,5].
%p A240139 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
%p A240139       abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
%p A240139       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
%p A240139     end:
%p A240139 a:= n-> b(n$2, -3):
%p A240139 seq(a(n), n=9..80);
%t A240139 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 A240139 a[n_] := b[n, n, -3];
%t A240139 a /@ Range[9, 80] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A240139 Column k=3 of A240021.
%K A240139 nonn
%O A240139 9,5
%A A240139 _Alois P. Heinz_, Apr 02 2014