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A331609 Number of compositions of n with the multiplicity of the first part even. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)