A331606 Number of compositions of n with the multiplicity of the first part odd.

1, 1, 4, 4, 12, 18, 44, 72, 158, 288, 604, 1146, 2332, 4528, 9126, 17944, 35940, 71130, 142132, 282344, 563630, 1121936, 2239060, 4462530, 8906236, 17764160, 35458774, 70761520, 141272876, 282025466, 563159588, 1124543256, 2245918406, 4485670168, 8960061076

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.

Formula

G.f.: Sum_{i>=1} (1-x)*x^i/(2*(-2*x^(i+1)+2*x^i-2*x+1)) + x/(2*(1-2*x)).

a(n) = A011782(n) - A331609(n). - Alois P. Heinz, Jan 23 2020

Example

For n=3, a(4)=4 as we count 4, 3+1, 1+3 and 2+1+1.

Maple

b:= proc(n, p, t) option remember; `if`(n=0, t,
      add(b(n-j, p, `if`(p=j, 1-t, t)), j=1..n))
    end:
a:= n-> add(b(n-j, j, 1), j=1..n):
seq(a(n), n=1..38);  # Alois P. Heinz, Jan 23 2020

Mathematica

gf[x_] := x/(2 (1 - 2 x)) + Sum[(1 - x) x^i/(2 (-2 x^(i + 1) + 2 x^i - 2 x + 1))  , {i, 1, 40}]; CL := CoefficientList[Series[gf[x], {x, 0, 35}], x];
Drop[CL, 1] (* Peter Luschny, Jan 23 2020 *)

Crossrefs

Cf. A011782, A331609 (similar, with even).

Sequence in context: A157617 A053415 A303315 * A079902 A309128 A120033

Adjacent sequences: A331603 A331604 A331605 * A331607 A331608 A331609

Keyword

nonn

Author

Arnold Knopfmacher, Jan 22 2020

Status

approved