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A048990 Catalan numbers with even index (A000108(2*n), n >= 0): a(n) = binomial(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

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.

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.

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

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Last modified January 20 10:23 EST 2018. Contains 297960 sequences.