|
%I M2804 #91 Mar 12 2021 22:32:36
%S 0,1,3,9,26,75,216,623,1800,5211,15115,43923,127854,372749,1088283,
%T 3181545,9312312,27287091,80038449,234988827,690513030,2030695569,
%U 5976418602,17601021837,51869858544,152951628725,451271872701,1332147482253
%N Number of directed animals of size n (k=1 column of A038622); number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, where s(0) = 2; also sum of row n+1 of array T in A026323.
%C Number of ordered trees with n+1 edges, having root degree at least 2 and nonroot outdegrees at most 2. - _Emeric Deutsch_, Aug 02 2002
%C From Petkovsek's algorithm, this recurrence does not have any closed form solutions. So there is no hypergeometric closed form for a(n). - Herbert S. Wilf
%C Sum of two consecutive trinomial coefficients starting two positions before central one. Example: a(4) = 10+16 and (1 + x + x^2)^4 = ... + 10*x^2 + 16*x^3 + 19*x^4 + ... - _David Callan_, Feb 07 2004
%C Image of n (A001477) under the Motzkin related matrix A107131. Binomial transform of A037952. - _Paul Barry_, May 12 2005
%C a(n) = total number of ascents (maximal runs of consecutive upsteps) in all Motzkin (n+1)-paths. For example, the 9 Motzkin 4-paths are FFFF, FFUD, FUDF, FUFD, UDFF, UDUD, UFDF, UFFD, UUDD and they contain a total of 9 ascents and so a(3)=9 (U=upstep, D=downstep, F=flatstep). - _David Callan_, Aug 16 2006
%C Image of the sequence (0,1,2,3,3,3,...) under the array A122896. - _Paul Barry_, Sep 18 2006
%C This is some kind of Motzkin transform of A079978 because the substitution x-> x*A001006(x) in the independent variable of the g.f. A079978(x) yields 1,0 followed by this sequence here. - _R. J. Mathar_, Nov 08 2008
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A005774/b005774.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H D. Gouyou-Beauchamps, G. Viennot, <a href="http://dx.doi.org/10.1016/0196-8858(88)90017-6">Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem</a>, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
%H Christian Krattenthaler, Daniel Yaqubi, <a href="https://arxiv.org/abs/1802.05990">Some determinants of path generating functions, II</a>, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de Séries Génératrices et Quelques Conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.
%H Simon Plouffe, <a href="http://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
%F Inverse binomial transform of [ 0, 1, 5, 21, 84, ... ] (A002054). - _John W. Layman_
%F D-finite with recurrence (n+2)*(n-1)*a(n) = 2*n*(n+1)*a(n-1) + 3*n*(n-1)*a(n-2) for all n in Z. - _Michael Somos_, May 01 2003
%F E.g.f.: exp(x)*(BesselI(1, 2*x)+BesselI(2, 2*x)). - _Vladeta Jovovic_, Jan 01 2004
%F G.f.: (1-x-sqrt(1-2x-3x^2))/(x(1-3x+sqrt(1-2x-3x^2))); a(n)= Sum_{k=0..n} C(k+1, n-k+1)*C(n, k)*k/(k+1); a(n) = Sum_{k=0..n} C(n, k)*C(k, floor((k-1)/2)). - _Paul Barry_, May 12 2005
%F Starting (1, 3, 9, 26, ...) = binomial transform of A026010: (1, 2, 4, 7, 14, 25, 50, 91, ...). - _Gary W. Adamson_, Oct 22 2007
%F a(n)*(2+n) = (4+4*n)*a(n-1) - n*a(n-2) + (12-6*n)*a(n-3). - _Simon Plouffe_, Feb 09 2012
%F a(n) ~ 3^(n+1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 10 2014
%F 0 = a(n)*(+36*a(n+1) + 18*a(n+2) - 96*a(n+3) + 30*a(n+4)) + a(n+1)*(-6*a(n+1) + 49*a(n+2) - 26*a(n+3) + 3*a(n+4)) + a(n+2)*(+15*a(n+3) - 8*a(n+4)) + a(n+3)*(a(n+4)) if n >= 0. - _Michael Somos_, Aug 06 2014
%F a(n) = GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2). - _Peter Luschny_, May 12 2016
%e G.f.: x + 3*x^2 + 9*x^3 + 26*x^4 + 75*x^5 + 216*x^6 + 623*x^7 + ...
%p seq( add(binomial(i,k+1)*binomial(i-k,k), k=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%p seq(simplify(GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2)), n=0..27); # _Peter Luschny_, May 12 2016
%t CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(x(1-3x+Sqrt[1-2x-3x^2])),{x,0,30}],x] (* _Harvey P. Dale_, Sep 20 2011 *)
%t RecurrenceTable[{a[0]==0, a[1]==1,a[n]==(2n(n+1)a[n-1]+3n(n-1)a[n-2])/ ((n+2)(n-1))},a,{n,30}] (* _Harvey P. Dale_, Nov 09 2012 *)
%o (PARI) s=[0,1]; {A005774(n)=k=(2*(n+2)*(n+1)*s[2]+3*(n+1)*n*s[1])/((n+3)*n); s[1]=s[2]; s[2]=k; k}
%o (PARI) {a(n) = if( n<2, n>0, (2 * (n+1) * n *a(n-1) + 3 * (n-1) * n * a(n-2)) / (n+2) / (n-1))}; /* _Michael Somos_, May 01 2003 */
%o (Haskell)
%o a005774 0 = 0
%o a005774 n = a038622 n 1 -- _Reinhard Zumkeller_, Feb 26 2013
%Y Cf. A038622, A098494, A026010, A005773, A005775, A066822.
%K nonn,easy,nice
%O 0,3
%A _Simon Plouffe_
%E Further descriptions from _Clark Kimberling_
|
