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A194797 Imbalance of the sum of parts of all partitions of n. 4
0, -2, 1, -7, 3, -21, 7, -49, 23, -97, 57, -195, 117, -359, 256, -624, 498, -1086, 909, -1831, 1634, -2986, 2833, -4847, 4728, -7700, 7798, -12026, 12537, -18633, 19745, -28479, 30723, -42955, 47100, -64284, 71136, -95228, 106402, -139718, 157327, -203495 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see example). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. Note that every column contains exactly the same parts. For more information see A135010.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{k=1..n} (-1)^(k-1)*k*(p(k)-p(k-1)), where p(k) is the number of partitions of k.

a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).

a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - Vaclav Kotesovec, Oct 09 2018

EXAMPLE

For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):

------------------------------------------------------

Partitions                Tree             Table 1.0

of 6.                    A194805            A135010

------------------------------------------------------

6                   6                     6 . . . . .

3+3                   3                   3 . . 3 . .

4+2                     4                 4 . . . 2 .

2+2+2                     2               2 . 2 . 2 .

5+1                         1   5         5 . . . . 1

3+2+1                       1 3           3 . . 2 . 1

4+1+1                   4   1             4 . . . 1 1

2+2+1+1                   2 1             2 . 2 . 1 1

3+1+1+1                     1 3           3 . . 1 1 1

2+1+1+1+1                 2 1             2 . 1 1 1 1

1+1+1+1+1+1                 1             1 1 1 1 1 1

------------------------------------------------------

.

.                   6 3 4 2 1 3 5

.     Table 2.0     . . . . 1 . .     Table 2.1

.      A182982      . . . 2 1 . .      A182983

.                   . 3 . . 1 2 .

.                   . . 2 2 1 . .

.                   . . . . 1

------------------------------------------------------

The sum of all parts > 1 on the left hand side is 34 and the sum of all parts > 1 on the right hand side is 13, so a(6) = -34 + 13 = -21. On the other hand for n = 6 the first n terms of A138880 are 0, 2, 3, 8, 10, 24 thus a(6) = 0-2+3-8+10-24 = -21.

MAPLE

with(combinat):

a:= proc(n) option remember;

      n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)

    end: a(0):=0:

seq(a(n), n=1..50); # Alois P. Heinz, Apr 04 2012

MATHEMATICA

a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* Jean-Fran├žois Alcover, Dec 09 2016 *)

CROSSREFS

Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.

Sequence in context: A258235 A330877 A021050 * A255138 A115629 A296461

Adjacent sequences:  A194794 A194795 A194796 * A194798 A194799 A194800

KEYWORD

sign

AUTHOR

Omar E. Pol, Jan 31 2012

STATUS

approved

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Last modified September 30 12:21 EDT 2020. Contains 337439 sequences. (Running on oeis4.)