A064310 Generalized Catalan numbers C(-1; n).

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

Wolfdieter Lang, Sep 21 2001

Keywords

• sign
• easy

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Unsigned sequence with a(0) := 0 is A000957 (Fine).

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.

Crossrefs

Cf. A000108, A000957, A039599, A106566.

Programs

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

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 *)

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

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)
A064310_list = list(islice(A064310_gen(),20)) # Chai Wah Wu, Apr 27 2023

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

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