login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1. 5
1, 1, 7, 33, 191, 1153, 7295, 47617, 318463, 2170881, 15028223, 105365505, 746651647, 5339185153, 38478839807, 279201841153, 2037998419967, 14954803494913, 110255315877887, 816299567480833, 6066679566041087 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

W. Lang, On polynomials related to derivatives of the generating function of Catalan numbers, Fib. Quart. 40,4 (2002) 299-313; Eq.(31) with lambda=-1/2.

FORMULA

a(n) = (-1)^n*Sum_{k=0,..,n} (C(k)/(-1/2)^k) with C(k)=A000108(k) (Catalan).

a(n) = -a(n-1) + C(n)*2^n, n >= 0, a(-1) := 0, with C(n)=A000108(n).

G.f.: A(2*x)/(1+x), with A(x) g.f. of Catalan numbers A000108.

Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Dec 09 2013

a(n) ~ 2^(3*n+3)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 09 2013

MATHEMATICA

CoefficientList[Series[(1-Sqrt[1-8*x])/(4*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 09 2013 *)

Table[FullSimplify[2^(n+1)*(2*n+2)! * Hypergeometric2F1Regularized[1, n+3/2, n+3, -8]/(n+1)! + (-1)^n/2], {n, 0, 20}] (* Vaclav Kotesovec, Dec 09 2013 *)

Table[(-1)^n*Sum[(-2)^k * CatalanNumber[k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)

PROG

(Sage)

def A064306():

    f, c, n = 1, 1, 1

    while True:

        yield f

        n += 1

        c = c * (8*n - 12) // n

        f = c - f

a = A064306()

print [a.next() for _ in range(21)] # Peter Luschny, Nov 30 2016

(PARI) for(n=0, 25, print1((-1)^n*sum(k=0, n, (-2)^k*binomial(2*k, k)/(k+1)), ", ")) \\ G. C. Greubel, Jan 27 2017

CROSSREFS

Sequence in context: A275860 A054256 A085636 * A292427 A215125 A204706

Adjacent sequences:  A064303 A064304 A064305 * A064307 A064308 A064309

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)