OFFSET
1,4
COMMENTS
Consider the three-dimensional structure of the shell model of partitions version "tree". Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns. Note that every column contains exactly the same parts, the same as a periodic table (see example). For more information see A135010.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = Sum_{k=1..n} (-1)^(k-1)*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*A002865(k).
a(n) ~ (-1)^(n+1) * Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Nov 11 2015
EXAMPLE
For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
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Partitions Tree Table 1.0
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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
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.
. 6 3 4 2 1 3 5
. Table 2.0 . . . . 1 . . Table 2.1
. . 3 . . 1 2 .
. . . 2 2 1 . .
. . . . . 1
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The number of partitions with parts on the left hand side is equal to 7 and the number of partitions with parts on the right hand side is equal to 3, so a(6) = -7+3 = -4. On the other hand; for n = 6 the first n terms of A002865 (with positive indices) are 0, 1, 1, 2, 2, 4 therefore a(6) = 0-1+1-2+2-4 = -4.
MAPLE
with(combinat):
a:= proc(n) option remember;
(-1)^n *(numbpart(n-1)-numbpart(n)) +`if`(n>1, a(n-1), 0)
end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012
MATHEMATICA
a[n_] := a[n] = (-1)^n*(PartitionsP[n-1]-PartitionsP[n]) + If[n>1, a[n-1], 0]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
nmax = 60; Rest[CoefficientList[Series[x/(1-x) - (1+x)/(1-x) * Product[1/((1 + x^(2*k-1))*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)
nmax = 60; Rest[CoefficientList[Series[-x/(1+x) - (1-x)/(1+x) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Omar E. Pol, Feb 02 2012
STATUS
approved