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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-Universität, 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 MATHEMATICA T[_, 0] = 1; T[1, 1] = 2; T[n_, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[_, _] = 0; a[n_] := T[n-1, n-1]; Array[a, 30] (* Jean-François Alcover, Jul 30 2018 *) 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 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 October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)