

A236634


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


2



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
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OFFSET

1,2


COMMENTS

Number of partitions of n whose rank is not 0.


LINKS



FORMULA



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]}];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



