A064333 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A064333

Generalized Catalan numbers C(-11; n).

1, 1, -10, 221, -6082, 187386, -6184848, 213843477, -7645509706, 280351640702, -10485617230780, 398467433529298, -15341431926699284, 597149747213056324, -23459916801814723548, 929028306450848244741, -37045540042729366580442

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..600

FORMULA

a(n) = Sum_{m-0..n-1} (n-m)*binomial(n-1+m, m)*(-11)^m/n.

a(n) = (1/12)^n*(1 + 11*Sum_{k=0..n-1} C(k)*(-11*12)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+11*x*c(-11*x)/12)/(1-x/12) = 1/(1-x*c(-11*x)) with c(x) g.f. of Catalan numbers A000108.

MATHEMATICA

CoefficientList[Series[(23 +Sqrt[1+44*x])/(2*(12-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((23 +sqrt(1+44*x))/(2*(12-x))) \\ G. C. Greubel, May 03 2019

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (23 +Sqrt(1+44*x))/(2*(12-x)) )); // G. C. Greubel, May 03 2019

(Sage) ((23 +sqrt(1+44*x))/(2*(12-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

CROSSREFS

Cf. A000108, A064334.

Sequence in context: A166181 A199748 A210137 * A223817 A317171 A229256

Adjacent sequences: A064330 A064331 A064332 * A064334 A064335 A064336

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved