A222044 Sum of smallest parts of all partitions of n into an odd number of parts.

0, 1, 2, 4, 5, 8, 11, 15, 19, 28, 35, 47, 61, 80, 102, 136, 168, 218, 276, 350, 437, 556, 686, 860, 1063, 1321, 1620, 2005, 2443, 2998, 3649, 4445, 5377, 6531, 7863, 9496, 11398, 13694, 16373, 19603, 23347, 27834, 33058, 39259, 46467, 55020, 64914, 76599

Offset: 0

Author

Alois P. Heinz, Feb 06 2013

Keywords

• nonn

Comments

a(n) + A222045(n) = A046746(n).

a(n) - A222045(n) = A222046(n).

Examples

			a(6) = 11: 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], sum of smallest parts is 1+2+1+1+6 = 11.
		

Links

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

Crossrefs

Cf. A046746, A211373, A222045, A222046, A222047, A222048, A222049.

Programs

Maple

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-> b(n, n)[1]:
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[nJean-François Alcover, Feb 03 2017, translated from Maple *)
Table[Total[Min/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* Harvey P. Dale, Jul 05 2019 *)

Formula

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