OFFSET
0,5
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [N. J. A. Sloane, Oct 08 2012]
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
P. Pagacz and M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv:1310.2122 [math.PR], 2013.
FORMULA
a(n) = Sum_{m=0..n-1} (-1)^m*(n-m)*binomial(n-1+m, m)/n.
a(n) = ((1/2)^n)*(1 + Sum_{k=0..n-1} C(k)*(-2)^k ), n >= 1, a(0)= 1, with C(n)=A000108(n) (Catalan).
G.f.: (1+x*c(-x)/2)/(1-x/2) = 1/(1-x*c(-x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{k=0..n} (-1)^(n-k)*A106566(n, k). - Philippe Deléham, Sep 18 2005
(-1)^n*a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k. - Philippe Deléham, Mar 13 2007
Conjecture: 2*n*a(n) + (7*n-12)*a(n-1) + 2*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Dec 02 2012
MATHEMATICA
a[n_]:= (1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 17 2013 *)
PROG
(PARI) {a(n) = (1 + sum(k=0, n-1, (-2)^k*binomial(2*k, k)/(k+1)))/2^n};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019
(Magma) [1] cat [(1 +(&+[(-2)^k*Binomial(2*k, k)/(k+1): k in [0..n-1]]))/2^n: n in [1..30]]; // G. C. Greubel, Feb 27 2019
(Sage) [1] + [(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1)))/2^n for n in (1..30)] # G. C. Greubel, Feb 27 2019
(Python)
from itertools import count, islice
def A064310_gen(): # generator of terms
yield from (1, 1, 0)
a, c = 0, 1
for n in count(1):
yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)*(1 if n&1 else -1)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Sep 21 2001
STATUS
approved