A194797 -id:A194797 - cp's OEIS frontend

A194795 Imbalance of the number of partitions of n.

0, -1, 0, -2, 0, -4, 0, -7, 1, -11, 3, -18, 6, -28, 13, -42, 24, -64, 41, -96, 69, -141, 112, -208, 175, -303, 271, -437, 410, -629, 609, -898, 896, -1271, 1302, -1792, 1868, -2510, 2660, -3493, 3752, -4839, 5248, -6666, 7293, -9131, 10065, -12454

Offset: 1

Omar E. Pol, Feb 02 2012

sign

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.

			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 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.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A002865, A135010, A182703, A187219, A194796, A194797, A194805, A194809.

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

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 *)

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) * (A240690(n+1) - A240690(n)) - 1. - Vaclav Kotesovec, Nov 11 2015
a(n) ~ (-1)^(n+1) * Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Nov 11 2015

A194796 Imbalance of the number of parts of all partitions of n.

Original entry on oeis.org

0, -1, 0, -3, 0, -8, 0, -17, 3, -31, 10, -58, 22, -101, 52, -167, 104, -278, 191, -451, 344, -711, 594, -1119, 983, -1730, 1606, -2635, 2555, -3990, 3978, -5972, 6118, -8835, 9269, -12986, 13835, -18917, 20454, -27320, 29900, -39204, 43268, -55846, 62112

Offset: 1

Omar E. Pol, Feb 01 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see the illustration in A194795). 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. For more information see A135010.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A006128, A096541, A135010, A138121, A138135, A138137, A197595, A194797, A194809.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= proc(n) option remember;
      (-1)^n*(b(n-1, n-1)[2]-b(n, n)[2])+`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 04 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := a[n] = (-1)^n*(b[n-1, n-1][[2]] - b[n, n][[2]]) + If[n == 1, 0, a[n-1]]; Table [a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

vector(50, n, sum(k=1, n, (-1)^(k-1)*(numdiv(k)-1+sum(j=1, k-1, (numdiv(j)-1)*(numbpart(k-j)-numbpart(k-j-1)))))) \\ Altug Alkan, Nov 11 2015

a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).

More terms from Alois P. Heinz, Apr 04 2012

A194809 Imbalance of the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, -2, 1, -5, 3, -12, 7, -25, 17, -47, 36, -88, 69, -155, 133, -262, 240, -439, 415, -717, 705, -1142, 1165, -1803, 1874, -2797, 2975, -4276, 4632, -6478, 7094, -9698, 10741, -14355, 16059, -21079, 23719, -30670, 34716, -44243, 50315, -63372

Offset: 1

Omar E. Pol, Feb 02 2012

sign

Consider the three-dimensional structure of the shell model of partitions version "tree". Note that only the larges parts > 1 produce the imbalance. Note that every column where is located a largest part contains largest parts of the same size, thesame as a periodic table (see example). For more information see A135010.

			For n = 6 the illustration of the shell model with 6 shells shows an imbalance of largest parts (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
------------------------------------------------------
The sum of largest parts > 1 on the left hand side is 23 and the sum of largest parts > 1 on the right hand side is 11, so a(6) = -23 + 11 = -12. On the other hand for n = 6 we have that 0 together with the first n-1 terms > 1 of A138137 are 0, 2, 3, 6, 8, 15 so a(6) = 0-2+3-6+8-15 = -12.

Cf. A135010, A138121, A138137, A141285, A194795-A194797, A194805.

a(n) = Sum_{k=2..n} (-1)^(k-1)*A138137(k), n >= 2.

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A194795 Imbalance of the number of partitions of n.

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A194796 Imbalance of the number of parts of all partitions of n.

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A194809 Imbalance of the sum of largest parts of all partitions of n.

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