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

Author

Arnold Knopfmacher, Jan 22 2020

Keywords

• nonn

Examples

			For n=4, a(4)=4 and counts 2+2, 1+2+1, 1+1+2 and 1+1+1+1.
		

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.

Crossrefs

Cf. A011782, A331606 (similar, with odd).

Programs

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 *)

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