A339435 G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

1, -1, -1, 1, 1, 3, -3, -1, -7, -11, 7, 3, 15, 35, 71, -35, 25, -57, -99, -277, -415, 25, -185, 39, 327, 1079, 1895, 3745, -71, 2907, 813, 479, -4927, -7259, -20393, -29877, -8409, -28621, -23041, -16811, 4441, 27783, 102741, 169595, 324065, 105265, 361471, 280983, 385215

Offset

0,6

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Formula

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

Maple

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, p!
      *(1-2*irem(p, 2)), add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 04 2020

Mathematica

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]

Crossrefs

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

Sequence in context: A114422 A127501 A181304 * A118408 A079268 A102316

Adjacent sequences: A339432 A339433 A339434 * A339436 A339437 A339438

Keyword

sign,look

Author

Ilya Gutkovskiy, Dec 04 2020

Status

approved