A240141 Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 5.

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 10, 1, 13, 2, 18, 4, 23, 7, 30, 12, 37, 19, 47, 30, 57, 44, 70, 64, 85, 90, 103, 125, 124, 169, 150, 227, 181, 298, 220, 388, 268, 498, 328, 634, 404, 797, 500, 996, 622, 1232, 775, 1515, 971, 1849, 1216, 2245, 1527, 2708

Offset

25,5

Comments

With offset 30 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -5.

Links

Alois P. Heinz, Table of n, a(n) for n = 25..1000

Formula

a(n) = [x^n y^5] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

Example

a(39) = 13: [23,7,5,3,1], [21,9,5,3,1], [19,11,5,3,1], [19,9,7,3,1], [17,13,5,3,1], [17,11,7,3,1], [17,9,7,5,1], [15,13,7,3,1], [15,11,9,3,1], [15,11,7,5,1], [15,9,7,5,3], [13,11,9,5,1], [13,11,7,5,3].
a(40) = 2: [13,9,7,5,3,2,1], [11,9,7,5,4,3,1].

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, -5):
seq(a(n), n=25..100);

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, -5];
a /@ Range[25, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

Crossrefs

Column k=5 of A240021.

Sequence in context: A362369 A320311 A240140 * A049641 A240142 A240143

Adjacent sequences: A240138 A240139 A240140 * A240142 A240143 A240144

Keyword

nonn

Author

Alois P. Heinz, Apr 02 2014

Status

approved