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

9,5

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.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 9..10000 (terms 9..1000 from Alois P. Heinz)

Formula

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

Example

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].

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 *)

Crossrefs

Column k=3 of A240021.

Sequence in context: A008735 A239241 A263395 * A392673 A390218 A360952

Adjacent sequences: A240136 A240137 A240138 * A240140 A240141 A240142

Keyword

nonn

Author

Alois P. Heinz, Apr 02 2014

Status

approved