A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts.

0, 2, 2, 4, 6, 10, 12, 20, 26, 38, 50, 70, 90, 124, 160, 212, 272, 356, 450, 582, 732, 932, 1166, 1470, 1824, 2280, 2814, 3486, 4280, 5268, 6428, 7864, 9552, 11614, 14044, 16990, 20450, 24626, 29524, 35392, 42272, 50472, 60060, 71444, 84734, 100432, 118736

Offset: 1

Author

Omar E. Pol, Feb 18 2014

Keywords

• nonn

Comments

Number of partitions of n whose rank is not 0.

Examples

			For n = 5 we have:
-------------------------------------------------------
Partitions    Largest    Number    Dyson's
of 5           part     of parts    rank       Type
-------------------------------------------------------
5                5    -    1    =    4      unbalanced
4+1              4    -    2    =    2      unbalanced
3+2              3    -    2    =    1      unbalanced
3+1+1            3    -    3    =    0      balanced
2+2+1            2    -    3    =   -1      unbalanced
2+1+1+1          2    -    4    =   -2      unbalanced
1+1+1+1+1        1    -    5    =   -4      unbalanced
-------------------------------------------------------
There are 6 partitions whose rank is not 0, so a(5) = 6.
		

Crossrefs

Cf. A000041, A047993, A064173, A209616.

Programs

Mathematica

P = PartitionsP;
a[n_] := P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}];
a /@ Range[46] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

Formula

a(n) = A000041(n) - A047993(n) = 2*A064173(n).