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A100320 A Catalan transform of (1+2x)/(1-2x). 12
1, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864, 5408312, 20801200, 80233200, 310235040, 1202160780, 4667212440, 18150270600, 70690527600, 275693057640, 1076515748880, 4208197927440, 16466861455200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A Catalan transform of (1+2x)/(1-2x) under the mapping g(x)->g(xc(x)). The original sequence can be retrieved by g(x)->g(x(1-x)).

T(2n,n) for the triangle A132046. Hankel transform is A144704. [From Paul Barry, Sep 19 2008]

Central terms of the triangle in A124927. [Reinhard Zumkeller, Mar 04 2012]

a(n) = A095660(2*n,n) for n > 0. [Reinhard Zumkeller, Apr 08 2012]

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

FORMULA

G.f.: (1+2xc(x))/(1-2xc(x)) where c(x) is the g.f. of A000108.

a(n) = 4*binomial(2n-1, n)-3*0^n.

a(n) = binomial(2n, n)(4*2^(n-1)-0^n)/2^n.

a(n) = sum{j=0..n, sum{k=0..n, C(2n, n-k)((2k+1)/(n+k+1))C(k, j)(-1)^(j-k)*(4*2^(j-1)-0^j)}}.

a(n)=A028329(n), n>0. [From R. J. Mathar, Sep 02 2008]

a(n)=Sum_{k, 0<=k<=n} A039599(n,k)*A010684(k). [From Philippe Deléham, Oct 29 2008]

G.f.: G(0) -1, where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

a(n) = [x^n] (1 + 2*x)/(1 - x)^(n+1). - Ilya Gutkovskiy, Oct 12 2017

PROG

(Haskell)

a100320 n = a124927 (2 * n) n  -- Reinhard Zumkeller, Mar 04 2012

CROSSREFS

Sequence in context: A102433 A221652 A259806 * A064649 A149332 A149333

Adjacent sequences:  A100317 A100318 A100319 * A100321 A100322 A100323

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 14 2004

EXTENSIONS

Incorrect connection with A046055 deleted by N. J. A. Sloane, Jul 08 2009

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.