A240140 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.

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, 49, 83, 59, 113, 73, 150, 89, 197, 112, 254, 141, 324, 180, 408, 231, 509, 298, 629, 386, 771, 500, 938, 648, 1135, 835, 1365, 1076, 1634, 1376, 1949, 1755, 2317

Offset

16,5

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 16..1000

Formula

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

Example

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

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, -4):
seq(a(n), n=16..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, -4];
a /@ Range[16, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A240021.

Sequence in context: A239968 A362369 A320311 * A240141 A049641 A240142

Adjacent sequences: A240137 A240138 A240139 * A240141 A240142 A240143

Keyword

nonn

Author

Alois P. Heinz, Apr 02 2014

Status

approved