A222045 Sum of smallest parts of all partitions of n into an even number of parts.

0, 0, 1, 1, 4, 4, 9, 10, 19, 21, 34, 40, 62, 72, 103, 124, 173, 207, 279, 337, 445, 538, 694, 842, 1077, 1299, 1634, 1977, 2464, 2969, 3669, 4411, 5410, 6488, 7896, 9447, 11442, 13640, 16421, 19536, 23411, 27761, 33124, 39174, 46554, 54915, 65008, 76485, 90258

Offset

0,5

Comments

A222044(n) + a(n) = A046746(n).

A222044(n) - a(n) = A222046(n).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 2019

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], sum of smallest parts is 1+1+1+3+2+1 = 9.

Maple

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-> b(n, n)[2]:
seq(a(n), n=0..60);

Mathematica

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_] := b[n, n][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Crossrefs

Cf. A046746, A211373, A222044, A222046, A222047, A222048, A222049.

Sequence in context: A192032 A116682 A168157 * A088190 A378032 A092322

Adjacent sequences: A222042 A222043 A222044 * A222046 A222047 A222048

Keyword

nonn

Author

Alois P. Heinz, Feb 06 2013

Status

approved