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A119259 Central terms of the triangle in A119258. 4
1, 3, 17, 111, 769, 5503, 40193, 297727, 2228225, 16807935, 127574017, 973168639, 7454392321, 57298911231, 441739706369, 3414246490111, 26447737520129, 205272288591871, 1595964714385409, 12427568655368191, 96905907580960769, 756583504975757311, 5913649000782757889 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, eq (42)

FORMULA

a(n) = A119258(2*n,n).

a(n) = sum{k=0..n, C(2n,k)*C(2n-k-1,n-k)};. - Paul Barry, Sep 28 2007

a(n) = sum{k=0..n, C(n+k-1,k)2^k}. - Paul Barry, Sep 28 2007

2*a(n) = A064062(n)+A178792(n). - Joseph Abate, Jul 21 2010

G.f.: (4*x^2+3*sqrt(1-8*x)*x-5*x)/(sqrt(1-8*x)*(2*x^2+x-1)-8*x^2-7*x+1). - Vladimir Kruchinin, Aug 19 2013

a(n) = (-1)^n - 2^(n+1)*binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2],2). - Peter Luschny, Jul 25 2014

a(n) = (-1)^n + 2^(n+1)*A014300(n). - Peter Luschny, Jul 25 2014

a(n) = [x^n] ( (1 + x)^2/(1 - x) )^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 67*x^3 + ... is essentially the o.g.f for A064602. - Peter Bala, Oct 01 2015

The o.g.f. is the diagonal of the bivariate rational function 1/(1 - t*(1 + x)^2/(1 - x)) and hence is algebraic by Stanley 1999, Theorem 6.33, p.197. - Peter Bala, Aug 21 2016

n*(3*n-4)*a(n) +(-21*n^2+40*n-12)*a(n-1) -4*(3*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

MATHEMATICA

Table[Binomial[2k - 1, k] Hypergeometric2F1[-2k, -k, 1 - 2k, -1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PROG

(Haskell)

a119259 n = a119258 (2 * n) n  -- Reinhard Zumkeller, Aug 06 2014

CROSSREFS

Cf. A005408, A056220, A000225, A000337, A055580, A027608, A119258, A064062, A178792, A014300, A064602.

Sequence in context: A074556 A295808 A215048 * A249921 A174404 A271611

Adjacent sequences:  A119256 A119257 A119258 * A119260 A119261 A119262

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, May 11 2006

STATUS

approved

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Last modified December 15 12:13 EST 2019. Contains 329999 sequences. (Running on oeis4.)