A331609 Number of compositions of n with the multiplicity of the first part even.

0, 1, 0, 4, 4, 14, 20, 56, 98, 224, 420, 902, 1764, 3664, 7258, 14824, 29596, 59942, 120012, 241944, 484946, 975216, 1955244, 3926078, 7870980, 15790272, 31650090, 63456208, 127162580, 254845446, 510582236, 1022940392, 2049048890, 4104264424, 8219808108

Offset

1,4

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

a(n) = A011782(n) - A331606(n). - Alois P. Heinz, Jan 23 2020

Example

For n=4, a(4)=4 and counts 2+2, 1+2+1, 1+1+2 and 1+1+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, 0), j=1..n):
seq(a(n), n=1..38);  # Alois P. Heinz, Jan 23 2020

Mathematica

gf[x_] := (1 - x)/(1 - 2 x) - Sum[ ((x - 1) x^i (-x^(i + 1) + x^i - 2 x + 1)) / ((2 x - 1) (-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, A331606 (similar, with odd).

Sequence in context: A263871 A326982 A263797 * A174406 A270844 A287286

Adjacent sequences: A331606 A331607 A331608 * A331610 A331611 A331612

Keyword

nonn

Author

Arnold Knopfmacher, Jan 22 2020

Status

approved