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

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

1,4

Comments

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.

Links

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

Formula

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

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A011081 A209491 A201575 * A147600 A022895 A197416

Adjacent sequences: A194793 A194794 A194795 * A194797 A194798 A194799

Keyword

sign

Author

Omar E. Pol, Feb 01 2012

Extensions

More terms from Alois P. Heinz, Apr 04 2012

Status

approved