A222046 Difference between sums of smallest parts of all partitions of n into odd number of parts and into even number of parts.

0, 1, 1, 3, 1, 4, 2, 5, 0, 7, 1, 7, -1, 8, -1, 12, -5, 11, -3, 13, -8, 18, -8, 18, -14, 22, -14, 28, -21, 29, -20, 34, -33, 43, -33, 49, -44, 54, -48, 67, -64, 73, -66, 85, -87, 105, -94, 114, -120, 132, -128, 156, -159, 174, -172, 203, -213, 234, -232, 263

Offset: 0

Author

Alois P. Heinz, Feb 06 2013

Keywords

• sign

Examples

			a(6) = 2 = (1+2+1+1+6) - (1+1+1+3+2+1) because the 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] and the 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].
		

Links

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

Crossrefs

Cf. A211373, A222044, A222045, 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-> (l->l[1]-l[2])(b(n, n)):
seq(a(n), n=0..100);

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, Jan 23 2017, translated from Maple *)

Formula

a(n) = A222044(n) - A222045(n).

a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 06 2019