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

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = C(4*n,2*n)/(2*n+1). 14
1, 2, 14, 132, 1430, 16796, 208012, 2674440, 35357670, 477638700, 6564120420, 91482563640, 1289904147324, 18367353072152, 263747951750360, 3814986502092304, 55534064877048198, 812944042149730764, 11959798385860453492, 176733862787006701400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

With interpolated zeros, this is C(n)(1+(-1)^n)/2 with g.f. given by 2/(sqrt(1+4x)+sqrt(1-4x)). - Paul Barry, Sep 09 2004

Self-convolution of a(n)/4^n gives Catalan numbers (A000108). - Vladimir Reshetnikov, Oct 10 2016

REFERENCES

Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

G. Markowsky, A method for deriving hypergeometric and related identities from the H^2 Hardy norm of conformal maps, arXiv preprint arXiv:1205.2458 [math.CV], 2012.

FORMULA

a(n) = 2 * A065097(n).

G.f.: A(x) = sqrt(1/8*x^-1*(1-sqrt(1-16*x))).

G.f.: 2F1( (1/4, 3/4); (3/2))(16*x). - Olivier Gérard Feb 17 2011

n*(2*n+1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Nov 30 2012

E.g.f: 2F2(1/4, 3/4; 1, 3/2; 16*x). - Vladimir Reshetnikov, Apr 24 2013

G.f. A(x) satisfies: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ). - Paul D. Hanna, Nov 09 2013

G.f. A(x) satisfies: A(x) = sqrt(1 + 4*x*A(x)^4). - Paul D. Hanna, Nov 09 2013

a(n) = hypergeom([1-2*n,-2*n],[2],1). - Peter Luschny, Sep 22 2014

a(n) ~ 2^(4*n-3/2)/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Oct 10 2016

EXAMPLE

sqrt(2*x^-1*(1-sqrt(1-x))) = 1 + 1/8*x + 7/128*x^2 + 33/1024*x^3 + ...

MATHEMATICA

f[n_] := CatalanNumber[ 2n]; Array[f, 18, 0] (* Or *)

CoefficientList[ Series[ Sqrt[2]/Sqrt[1 + Sqrt[1 - 16 x]], {x, 0, 17}], x] (* Robert G. Wilson v *)

CatalanNumber[Range[0, 40, 2]] (* Harvey P. Dale, Mar 19 2015 *)

PROG

(Mupad) combinat::dyckWords::count(2*n) $ n = 0..28 // Zerinvary Lajos, Apr 14 2007

(PARI) /* G.f.: A(x) = exp( x*A(x)^4 + Integral(A(x)^4 dx) ): */

{a(n)=local(A=1+x); for(i=1, n, A=exp(x*A^4 + intformal(A^4 +x*O(x^n)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 09 2013

for(n=0, 30, print1(a(n), ", "))

(Sage)

A048990 = lambda n: hypergeometric([1-2*n, -2*n], [2], 1)

[Integer(A048990(n).n()) for n in range(20)] # Peter Luschny, Sep 22 2014

CROSSREFS

Cf. A000108, A024492, A065097.

Sequence in context: A235352 A146971 A246481 * A089602 A052641 A157085

Adjacent sequences:  A048987 A048988 A048989 * A048991 A048992 A048993

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

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 December 11 15:43 EST 2017. Contains 295905 sequences.