A339435 - OEIS

%I #8 Dec 05 2020 21:14:52

%S 1,-1,-1,1,1,3,-3,-1,-7,-11,7,3,15,35,71,-35,25,-57,-99,-277,-415,25,

%T -185,39,327,1079,1895,3745,-71,2907,813,479,-4927,-7259,-20393,

%U -29877,-8409,-28621,-23041,-16811,4441,27783,102741,169595,324065,105265,361471,280983,385215

%N G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

%C The difference between the number of compositions (ordered partitions) of n into an even number of distinct parts and the number of compositions (ordered partitions) of n into an odd number of distinct parts.

%H Alois P. Heinz, <a href="/A339435/b339435.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) = A332305(n) - A332304(n).

%p b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, p!

%p       *(1-2*irem(p, 2)), add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 04 2020

%t nmax = 48; CoefficientList[Series[Sum[(-1)^k k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A010815, A032020, A081362, A332304, A332305.

%K sign,look

%O 0,6

%A _Ilya Gutkovskiy_, Dec 04 2020