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

0,5

Links

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

Formula

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

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

Example

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.

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<i, [0, 0], (l-> [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[n<i, {0, 0}, Reverse[b[n-i, i]]]]; a[n_] := g[n, 1][[2]] - b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

Crossrefs

Cf. A116686, A211870, A222045, A222048.

Sequence in context: A097701 A362548 A225596 * A056870 A326507 A014739

Adjacent sequences: A211878 A211879 A211880 * A211882 A211883 A211884

Keyword

nonn

Author

Alois P. Heinz, Feb 13 2013

Status

approved