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A100320 A Catalan transform of (1 + 2*x)/(1 - 2*x). 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 + 2*x)/(1 - 2*x) under the mapping g(x) -> g(x*c(x)). (Here c(x) is the g.f. of A000108.) The original sequence can be retrieved by g(x) -> g(x*(1-x)).

Hankel transform is A144704. - Paul Barry, Sep 19 2008

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

LINKS

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

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

FORMULA

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

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

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

a(n) = T(2*n,n), where T(n,k) = A132046(n,k). - Paul Barry, Sep 19 2008

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

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

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

MATHEMATICA

a[0] = 1; a[n_] := 2 Binomial[2 n, n];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 31 2018 *)

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 August 11 00:25 EDT 2020. Contains 336403 sequences. (Running on oeis4.)