A103420 Number of compositions of n in which the least part is even.

0, 1, 0, 2, 2, 4, 5, 11, 17, 28, 44, 75, 123, 203, 330, 541, 883, 1444, 2357, 3848, 6271, 10214, 16624, 27051, 43995, 71523, 116223, 188790, 306554, 497624, 807553, 1310177, 2125126, 3446237, 5587517, 9057611, 14680337, 23789891, 38546834, 62449682, 101163024

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Robert Israel)

Formula

G.f.: Sum((1-x)^2*x^(2*n)/((1-x-x^(2*n))*(1-x-x^(2*n+1))), n=1..infinity).

G.f.: Sum(x^(2*n)/((1-x)^n*(1+x^n)),n=1..infinity). - Vladeta Jovovic, Mar 02 2008

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-1). - Vaclav Kotesovec, May 01 2014

Maple

N:= 50: # for a(1) .. a(N)
G:= add(x^(2*n)/((1-x)^n*(1+x^n)), n=1..N/2):
S:= series(G, x, N+1):
[seq(coeff(S, x, i), i=1..N)]; # Robert Israel, Oct 23 2024
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1-
      irem(m, 2), add(b(n-j, min(m, j)), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..42);  # Alois P. Heinz, Oct 23 2024

Mathematica

Rest[ CoefficientList[ Series[ Expand[ Sum[(1 - x)^2*x^(2n)/((1 - x - x^(2n))*(1 - x - x^(2n + 1))), {n, 40}]], {x, 0, 40}], x]] (* Robert G. Wilson v, Feb 05 2005 *)

Crossrefs

Cf. A103419, A103421, A103422, A027187, A027193, A026804, A026805.

Sequence in context: A338048 A127825 A185100 * A329701 A032258 A153949

Adjacent sequences: A103417 A103418 A103419 * A103421 A103422 A103423

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 04 2005

Extensions

More terms from Robert G. Wilson v, Feb 05 2005

Status

approved