OFFSET
1,4
COMMENTS
Also number of partitions of n in which the largest part occurs an even number of times. Example: a(6)=3 because we have [3,3],[2,2,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
FORMULA
From Vladeta Jovovic, Aug 26 2003: (Start)
G.f.: Sum_{k>=2} ((-1)^k*(-1+1/Product_{i>=k} (1-x^i))).
From Emeric Deutsch, Apr 04 2006: (Start)
G.f.: Sum_{k>=1}(x^(2k)/Product_{j>=2k}(1-x^j)).
G.f.: Sum_{k>=1}(x^(2k)/((1-x^(2k))*Product_{j=1..k-1}(1-x^j))). (End)
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 61*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jul 06 2019, extended Nov 02 2019
EXAMPLE
a(6)=3 because we have [6],[4,2] and [2,2,2].
MAPLE
g:=sum(x^(2*k)/(1-x^(2*k))/product(1-x^j, j=1..k-1), k=1..40): gser:=series(g, x=0, 52): seq(coeff(gser, x, n), n=1..49); # Emeric Deutsch, Apr 04 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+
`if`(n=i, 1-irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=1..60); # Alois P. Heinz, Jul 26 2015
MATHEMATICA
b[n_, i_] := b[n, i] = If[n<1 || i<1, 0, b[n, i-1] + If[n==i, 1-Mod[n, 2], 0] + If[i>n, 0, b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved