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A025565 a(n) = T(n,n-1), where T is array defined in A025564. 13
1, 2, 4, 10, 26, 70, 192, 534, 1500, 4246, 12092, 34606, 99442, 286730, 829168, 2403834, 6984234, 20331558, 59287740, 173149662, 506376222, 1482730098, 4346486256, 12754363650, 37461564504, 110125172682, 323990062452, 953883382354 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n+1)=number of UDU-free paths of n upsteps (U) and n downsteps (D), n>=0. - David Callan, Aug 19 2004

Hankel transform is A120580. - Paul Barry, Mar 26 2010

If interpreted with offset 0, the inverse binomial transform of A006134 - Gary W. Adamson, Nov 10 2007

Also the number of different integer sets { k_1, k_2, ..., k_(i+1) } with sum_(j=1)^(i+1) k_j = i and k_j >= 0, see the "central binomial coefficients" (A000984), without all sets in which any two successive k_j and k_(j+1) are zero.  See the partition problem eq. 3.12 on p. 19 in my dissertation below. - Eva Kalinowski, Oct 18 2012

REFERENCES

Baccherini, D.; Merlini, D.; Sprugnoli, R. Binary words excluding a pattern and proper Riordan arrays. Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See p. 1034. - N. J. A. Sloane, Mar 25 2014

Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universit├Ąt, 2002.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014

FORMULA

G.f.: x*sqrt((1+x)/(1-3*x)).

a(n) = 2*A005773(n-1) for n>1.

a(n) = |A085455(n-1)| = A025577(n)-A025577(n-1) = A002426(n)+A002426(n-1).

Sum(Sum (-1)^(n-i)a(j)a(i-j), (j=0, .., i)), (i=0, .., n)=3^n - Mario Catalani (mario.catalani(AT)unito.it), Jul 02 2003

a(1) = 1, a(n) = M(n-1) + sum (M(k-1)*a(n-k), k=1..n-1) with M=A001006, the Motzkin Numbers. - Reinhard Zumkeller, Mar 30 2012

Conjecture: (-n+1)*a(n) +2*(n-1)*a(n-1) +3*(n-3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012

G.f.: G(0), where G(k)= 1 + 4*x*(4*k+1)/( (1+x)*(4*k+2) - x*(1+x)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1+x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013

a(n) = n*hypergeom([2-n, 1/2-n/2, 1-n/2], [2, -n], 4). - Peter Luschny, Jul 12 2016

a(n) = (-1)^n*2*hypergeom([3/2, 2-n], [2], 4) for n>1. - Peter Luschny, Jan 30 2017

EXAMPLE

G.f. = x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 70*x^6 + 192*x^7 + 534*x^8 + ...

MAPLE

seq( add(binomial(i-2, k)*(binomial(i-k, k+1)), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

# Alternatively:

a := n -> `if`(n=1, 1, 2*(-1)^n*hypergeom([3/2, 2-n], [2], 4)):

seq(simplify(a(n)), n=1..28); # Peter Luschny, Jan 30 2017

PROG

(Haskell)

a025565 n = a025565_list !! (n-1)

a025565_list = 1 : f a001006_list [1] where

   f (x:xs) ys = y : f xs (y : ys) where

     y = x + sum (zipWith (*) a001006_list ys)

-- Reinhard Zumkeller, Mar 30 2012

(Sage)

def A():

    a, b, n  = 1, 1, 1

    yield a

    while True:

        yield a + b

        n += 1

        a, b = b, ((3*(n-1))*a+(2*n-1)*b)//n

A025565 = A()

print([A025565.next() for _ in range(28)]) # Peter Luschny, Jan 30 2017

CROSSREFS

Cf. A025564.

First column of A097692.

Sequence in context: A113337 A084575 A081881 * A085455 A055226 A097085

Adjacent sequences:  A025562 A025563 A025564 * A025566 A025567 A025568

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Removed incorrect statement related to A000984 (see A002426) and duplicate of the g.f., R. J. Mathar, Oct 16 2009

Edited by R. J. Mathar, Aug 09 2010

STATUS

approved

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Last modified February 22 19:36 EST 2018. Contains 299469 sequences. (Running on oeis4.)