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A064310 Generalized Catalan numbers C(-1; n). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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).

REFERENCES

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012

LINKS

Table of n, a(n) for n=0..25.

S. B. Ekhad, 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, M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv preprint arXiv:1310.2122, 2013

FORMULA

a(n)= sum((n-m)*binomial(n-1+m, m)*((-1)^m)/n, m=0..n-1) = ((1/2)^n)*(1+sum(C(k)*(-2)^k, k=0..n-1)), 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<=k<=n} (-1)^(n-k)*A106566(n, k) . - Philippe Deléham, Sep 18 2005

(-1)^n*a(n)=Sum_{k, 0<=k<=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, 25}] (* Jean-François Alcover, Jul 17 2013 *)

CROSSREFS

Sequence in context: A125306 A209797 * A126983 A104629 A000957 A125305

Adjacent sequences:  A064307 A064308 A064309 * A064311 A064312 A064313

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved

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Last modified December 13 22:42 EST 2018. Contains 318087 sequences. (Running on oeis4.)