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%I #88 Jun 25 2022 21:42:47
%S 1,2,4,10,26,70,192,534,1500,4246,12092,34606,99442,286730,829168,
%T 2403834,6984234,20331558,59287740,173149662,506376222,1482730098,
%U 4346486256,12754363650,37461564504,110125172682,323990062452,953883382354
%N a(n) = T(n,n-1), where T is array defined in A025564.
%C a(n+1) is the number of UDU-free paths of n upsteps (U) and n downsteps (D), n>=0. - _David Callan_, Aug 19 2004
%C Hankel transform is A120580. - _Paul Barry_, Mar 26 2010
%C If interpreted with offset 0, the inverse binomial transform of A006134 - _Gary W. Adamson_, Nov 10 2007
%C 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
%H Reinhard Zumkeller, <a href="/A025565/b025565.txt">Table of n, a(n) for n = 1..1000</a>
%H Kassie Archer and Christina Graves, <a href="https://arxiv.org/abs/2205.09686">A new statistic on Dyck paths for counting 3-dimensional Catalan words</a>, arXiv:2205.09686 [math.CO], 2022.
%H Andrei Asinowski and Cyril Banderier, <a href="https://doi.org/10.4230/LIPIcs.AofA.2020.1">On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution</a>, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
%H D. Baccherini, D. Merlini, and R. Sprugnoli, <a href="http://dx.doi.org/10.1016/j.disc.2006.07.023">Binary words excluding a pattern and proper Riordan arrays</a>, Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003). See page 1034. - _N. J. A. Sloane_, Mar 25 2014
%H J. L. Jacobsen and J. Salas, <a href="http://arxiv.org/abs/cond-mat/0407444">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions</a>, J. Stat. Phys. 122 (2006) 705-760; arXiv:cond-mat/0407444, 2004-2006. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
%H Eva Kalinowski, <a href="https://doi.org/10.17192/z2002.0093">Mott-Hubbard-Isolator in hoher Dimension</a>, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
%F G.f.: x*sqrt((1+x)/(1-3*x)).
%F a(n) = 2*A005773(n-1) for n > 1.
%F a(n) = |A085455(n-1)| = A025577(n) - A025577(n-1) = A002426(n) + A002426(n-1).
%F Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*a(j)*a(i-j) = 3^n. - Mario Catalani (mario.catalani(AT)unito.it), Jul 02 2003
%F a(1) = 1, a(n) = M(n-1) + Sum_{k=1..n-1} M(k-1)*a(n-k) with M=A001006, the Motzkin Numbers. - _Reinhard Zumkeller_, Mar 30 2012
%F D-finite with recurrence: (-n+1)*a(n) +2*(n-1)*a(n-1) +3*(n-3)*a(n-2)=0. - _R. J. Mathar_, Dec 02 2012
%F 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
%F a(n) = n*hypergeom([2-n, 1/2-n/2, 1-n/2], [2, -n], 4). - _Peter Luschny_, Jul 12 2016
%F a(n) = (-1)^n*2*hypergeom([3/2, 2-n], [2], 4) for n > 1. - _Peter Luschny_, Jan 30 2017
%e G.f. = x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 70*x^6 + 192*x^7 + 534*x^8 + ...
%p 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
%p # Alternatively:
%p a := n -> `if`(n=1,1,2*(-1)^n*hypergeom([3/2, 2-n], [2], 4)):
%p seq(simplify(a(n)),n=1..28); # _Peter Luschny_, Jan 30 2017
%t 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;
%t a[n_] := T[n-1, n-1];
%t Array[a, 30] (* _Jean-François Alcover_, Jul 30 2018 *)
%o (Haskell)
%o a025565 n = a025565_list !! (n-1)
%o a025565_list = 1 : f a001006_list [1] where
%o f (x:xs) ys = y : f xs (y : ys) where
%o y = x + sum (zipWith (*) a001006_list ys)
%o -- _Reinhard Zumkeller_, Mar 30 2012
%o (Sage)
%o def A():
%o a, b, n = 1, 1, 1
%o yield a
%o while True:
%o yield a + b
%o n += 1
%o a, b = b, ((3*(n-1))*a+(2*n-1)*b)//n
%o A025565 = A()
%o print([next(A025565) for _ in range(28)]) # _Peter Luschny_, Jan 30 2017
%Y Cf. A005773, A025564.
%Y First column of A097692.
%Y Partial sums of A105696.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Incorrect statement related to A000984 (see A002426) and duplicate of the g.f. removed by _R. J. Mathar_, Oct 16 2009
%E Edited by _R. J. Mathar_, Aug 09 2010
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