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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
Programs
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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 *) -
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
Formula
a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).
Extensions
More terms from Alois P. Heinz, Apr 04 2012
Comments