A332304 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is odd.

0, 1, 1, 1, 1, 1, 7, 7, 13, 19, 25, 31, 43, 49, 61, 193, 205, 337, 475, 727, 985, 1363, 1741, 2359, 2983, 3841, 4705, 5929, 12193, 13777, 20527, 27631, 39901, 52651, 75601, 99151, 132907, 172297, 227053, 287569, 373525, 465241, 587563, 725839, 899761, 1457683

Offset

0,7

Links

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

Index entries for sequences related to compositions

Formula

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

a(n) = A032020(n) - A332305(n).

Example

a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Maple

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
      irem(p, 2)*p!, 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, Feb 09 2020

Mathematica

nmax = 45; CoefficientList[Series[Sum[(2 k - 1)! x^(k (2 k - 1))/Product[1 - x^j, {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A027193, A032020, A067659, A166444, A332305.

Sequence in context: A335895 A072821 A038589 * A317790 A109539 A109541

Adjacent sequences: A332301 A332302 A332303 * A332305 A332306 A332307

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 09 2020

Status

approved