A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.

1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156

Offset: 0

Author

Philippe Deléham, Mar 21 2007

Keywords

• sign

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.

Crossrefs

Cf. A000108, A000957, A039599, A064310, A106566.

Programs

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

Mathematica

Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)

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

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)
A126983_list = list(islice(A126983_gen(),20)) # Chai Wah Wu, Apr 27 2023

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

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