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

1,2

Comments

Number of partitions of n whose rank is not 0.

Links

Table of n, a(n) for n=1..47.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000041, A047993, A064173, A209616.

Sequence in context: A326528 A326633 A323587 * A034406 A098330 A240310

Adjacent sequences: A236631 A236632 A236633 * A236635 A236636 A236637

Keyword

nonn

Author

Omar E. Pol, Feb 18 2014

Status

approved