A211870 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an odd number of parts.

0, 0, 0, 0, 1, 3, 6, 13, 22, 38, 58, 93, 134, 202, 282, 405, 554, 774, 1035, 1412, 1862, 2489, 3243, 4267, 5496, 7137, 9106, 11684, 14782, 18782, 23575, 29689, 37010, 46238, 57275, 71048, 87489, 107844, 132083, 161853, 197243, 240418, 291619, 353702, 427167

Offset

0,6

Links

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

Formula

a(n) = A222047(n) - A222044(n).

a(n) = A116686(n) - A211881(n).

Example

a(6) = 6: 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], difference between sum of largest parts and sum of smallest parts is (2+2+3+4+6) - (1+2+1+1+6) = 17 - 11 = 6.

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)[1] -b(n, n)[1]:
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][[1]] - b[n, n][[1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)

Crossrefs

Cf. A116686, A211881, A222044, A222047.

Sequence in context: A280029 A178097 A361276 * A239987 A048134 A058397

Adjacent sequences: A211867 A211868 A211869 * A211871 A211872 A211873

Keyword

nonn

Author

Alois P. Heinz, Feb 12 2013

Status

approved