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A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts. 0
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 27 09:08 EDT 2020. Contains 338035 sequences. (Running on oeis4.)