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A126087 Expansion of c(2x^2)/(1-xc(2x^2)), where c(x) = (1-sqrt(1-4x))/(2x) is the g.f. of the Catalan numbers (A000108). 8
1, 1, 3, 5, 15, 29, 87, 181, 543, 1181, 3543, 7941, 23823, 54573, 163719, 381333, 1143999, 2699837, 8099511, 19319845, 57959535, 139480397, 418441191, 1014536117, 3043608351, 7426790749, 22280372247, 54669443141, 164008329423 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Series reversion of x(1+x)/(1+2x+3x^2) [offset 0]. - Paul Barry, Mar 13 2007

Hankel transform is 2^C(n+1,2). - Philippe Deléham, Mar 16 2007

LINKS

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

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

FORMULA

G.f.: [1-sqrt(1-8x^2)]/[x(4x-1+sqrt(1-8x^2))]. - Emeric Deutsch, Mar 04 2007

a(n) = Sum_{k, 0<=k<=n} 2^(n-k)*A120730(n,k). - Philippe Deléham, Oct 16 2008

a(n) = sum(k=1..n,(1+(-1)^(n-k))*k*2^((n-k)/2-1)*C(n,(n+k)/2)/n), n>0 [Vladimir Kruchinin, Feb 18 2011]

a(2n) = A089022(n). - Philippe Deléham, Nov 02 2011

Conjecture: (n+1)*a(n) -3*(n+1)*a(n-1) +8*(2-n)*a(n-2) +24*(n-2)*a(n-3) =0. - R. J. Mathar, Nov 14 2011

a(n) ~ 2^(3*(n+1)/2) * (3+2*sqrt(2) + (3-2*sqrt(2))*(-1)^n) / (n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Feb 13 2014

MAPLE

c:=x->(1-sqrt(1-4*x))/2/x: G:=c(2*x^2)/(1-x*c(2*x^2)): Gser:=series(G, x=0, 35): seq(coeff(Gser, x, n), n=0..32); # Emeric Deutsch, Mar 04 2007

MATHEMATICA

CoefficientList[Series[(1-Sqrt[1-8*x^2])/(x*(4*x-1+Sqrt[1-8*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

CROSSREFS

Cf. A000108.

Sequence in context: A166956 A048738 A018454 * A148498 A259921 A292689

Adjacent sequences:  A126084 A126085 A126086 * A126088 A126089 A126090

KEYWORD

nonn

AUTHOR

Philippe Deléham, Mar 03 2007

EXTENSIONS

More terms from Emeric Deutsch, Mar 04 2007

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.