A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.

0, 0, 0, 1, 2, 5, 9, 16, 26, 41, 65, 95, 142, 202, 293, 403, 568, 766, 1054, 1399, 1886, 2469, 3276, 4237, 5538, 7094, 9162, 11628, 14856, 18704, 23670, 29590, 37130, 46109, 57428, 70885, 87685, 107634, 132324, 161595, 197545, 240091, 291990, 353302, 427624

Offset: 0

Author

Alois P. Heinz, Feb 13 2013

Keywords

• nonn

Examples

			a(6) = 9: 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], difference between sum of largest parts and sum of smallest parts is (1+2+3+3+4+5) - (1+1+1+3+2+1) = 18 - 9 = 9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A116686, A211870, A222045, A222048.

Programs

Maple

g:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
      `if`(i>n, [0, 0], g(n, i+1)+(l-> [l[2], l[1]])(g(n-i, i)))
    end:
b:= proc(n, i) option remember;
      [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
       `if`(n [l[2], l[1]])(b(n-i, i))))
    end:
a:= n-> g(n, 1)[2] -b(n, n)[2]:
seq(a(n), n=0..50);

Mathematica

g[n_, i_] := g[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, g[n, i+1] + Reverse[g[n-i, i]]]; b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 15 2017, translated from Maple *)

Formula

a(n) = A222048(n) - A222045(n).

a(n) = A116686(n) - A211870(n).