A222049 Difference between sums of largest parts of all partitions of n into odd number of parts and into even number of parts.

0, 1, 1, 2, 0, 2, -1, 2, -4, 4, -6, 5, -9, 8, -12, 14, -19, 19, -22, 26, -32, 38, -41, 48, -56, 65, -70, 84, -95, 107, -115, 133, -153, 172, -186, 212, -240, 264, -289, 325, -366, 400, -437, 485, -544, 597, -649, 714, -799, 869, -942, 1037, -1148, 1246, -1351

Offset: 0

Author

Alois P. Heinz, Feb 06 2013

Keywords

• sign

Examples

			a(6) = -1 = (2+2+3+4+6) - (1+2+3+3+4+5) because the partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6] and the partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..6000

Crossrefs

Cf. A211373, A222044, A222045, A222046, A222047, A222048.

Programs

Maple

b:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
      `if`(i>n, [0, 0], b(n, i+1)+(l-> [l[2], l[1]])(b(n-i, i)))
    end:
a:= n-> (l->l[1]-l[2])(b(n, 1)):
seq(a(n), n=0..60);

Mathematica

b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, b[n, i+1] + Reverse @ b[n-i, i]]; a[n_] :=  b[n, 1][[1]]-b[n, 1][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

Formula

a(n) = A222047(n) - A222048(n).

G.f.: Sum_{i>=0} i*x^i/Product_{j=1..i} (1 + x^j). - Ilya Gutkovskiy, Apr 13 2018