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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Number of partitions of n whose rank is not 0.
LINKS
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
Sequence in context: A326528 A326633 A323587 * A034406 A098330 A240310
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 18 2014
STATUS
approved

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Last modified May 26 21:41 EDT 2024. Contains 372847 sequences. (Running on oeis4.)