OFFSET
0,5
COMMENTS
Hankel transform is (-1)^n.
Catalan transform of A033999. - R. J. Mathar, Nov 11 2008
LINKS
Fung Lam, Table of n, a(n) for n = 0..1500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
FORMULA
a(n) = (-1)^n*A064310(n).
a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
From Philippe Deléham, Nov 15 2009: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
a(n) = -A000957(n) for n>0. (End)
Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014
a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014
MATHEMATICA
Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)
PROG
(PARI) {a(n) = (-1/2)^n*(1+sum(k=0, n-1, (-2)^k*binomial(2*k, k)/(k+1)))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019
(Magma) [1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k, k)/(k+1): k in [0..n-1]])): n in [1..30]]; // G. C. Greubel, Feb 27 2019
(Sage) [1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # G. C. Greubel, Feb 27 2019
(Python)
from itertools import count, islice
def A126983_gen(): # generator of terms
yield from (1, -1, 0)
a, c = 0, 1
for n in count(1):
yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
CROSSREFS
KEYWORD
sign
AUTHOR
Philippe Deléham, Mar 21 2007
STATUS
approved