OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 06 2019
EXAMPLE
a(6) = 2 = (1+2+1+1+6) - (1+1+1+3+2+1) because the partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6] and the partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1].
MAPLE
b:= proc(n, i) option remember;
[`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
`if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
end:
a:= n-> (l->l[1]-l[2])(b(n, n)):
seq(a(n), n=0..100);
MATHEMATICA
b[n_, i_] := b[n, i] = {If[n == i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[n<i, {0, 0}, Reverse[b[n-i, i]]]]; a[n_] := (#[[1]] - #[[2]]&)[b[n, n] ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 23 2017, translated from Maple *)
CROSSREFS
KEYWORD
sign
AUTHOR
Alois P. Heinz, Feb 06 2013
STATUS
approved