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.

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, 33, 94, 44, 121, 58, 155, 79, 194, 107, 241, 146, 296, 197, 361, 267, 436, 355, 525, 472, 628, 618, 750, 805, 894, 1035, 1064, 1324, 1267, 1673, 1511, 2103, 1804

Offset: 9

Author

Alois P. Heinz, Apr 02 2014

Keywords

• nonn

Comments

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.

Examples

			a(20) = 2: [9,5,3,2,1], [7,5,4,3,1].
a(21) = 7: [17,3,1], [15,5,1], [13,7,1], [13,5,3], [11,9,1], [11,7,3], [9,7,5].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..1000

Crossrefs

Column k=3 of A240021.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, -3):
seq(a(n), n=9..80);

Mathematica

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]]]];
a[n_] := b[n, n, -3];
a /@ Range[9, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^3] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).