OFFSET
0,4
COMMENTS
Powers of 2 with additional first two terms.
[(-1)^n*a(n)] = [1, 0, 1, -2, 4, -8, 16, -32, ...] is the inverse binomial transform of A008619 = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...]. - Philippe Deléham, Nov 15 2009
Number of compositions (ordered partitions) of n into an even number of parts. - Geoffrey Critzer, Mar 28 2010
Number of compositions of n into an even number of even parts.
Number of compositions of n into parts k >= 2 where there are k - 1 sorts of part k. - Joerg Arndt, Sep 30 2012
Taking n-th differences of this sequence reproduces the same sequence except for a(1) = n mod 2 (parity of n) and a(0) = (-1)^a(1)*floor(n/2 + 1). - M. F. Hasler, Jan 13 2015
REFERENCES
Richard P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, 1997, p. 45, exercise 9.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Index entries for linear recurrences with constant coefficients, signature (2).
FORMULA
a(n) = 2^(n-2), n >= 2; a(0) = 1, a(1) = 0.
G.f.: (1-x)^2/(1-2*x).
G.f. 1/( 1 - Sum_{k >= 1} (k-1)*x^k ). - Joerg Arndt, Sep 30 2012
G.f.: x*G(0), where G(k) = 1 + 1/(1 - (1 - x)/(1 + x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
Inverse binomial transform of (3^n - 2*n + 1)/2 for n >= 0. - Paul Curtz, Sep 24 2019
E.g.f.: (1/4)*(3 + exp(2*x) - 2*x). - Stefano Spezia, Sep 25 2019
MAPLE
MATHEMATICA
a = x/(1 - x); CoefficientList[Series[1/(1 - a^2), {x, 0, 30}], x] (* Geoffrey Critzer, Mar 28 2010 *)
PROG
(PARI) a(n)=if(n<2, n==0, 2^(n-2))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
Additional comments from Barry E. Williams, May 27 2000
Additional comments from Michael Somos, Jun 18 2002
Edited by M. F. Hasler, Jan 13 2015
STATUS
approved