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%I M2748 N1104 #139 Feb 02 2024 19:46:38
%S 1,1,3,8,24,75,243,808,2742,9458,33062,116868,417022,1500159,5434563,
%T 19808976,72596742,267343374,988779258,3671302176,13679542632,
%U 51134644014,191703766638,720629997168,2715610275804,10256844598900,38822029694628,147229736485868
%N Number of ordered rooted trees with n edges having root of odd degree.
%C a(n) is the number of Dyck n-paths containing no peak at height 2 before the first return to ground level. Example: a(3)=3 counts UUUDDD, UDUUDD, UDUDUD. - _David Callan_, Jun 07 2006
%C Also number of order trees with n edges and having no even-length branches starting at the root. - _Emeric Deutsch_, Mar 02 2007
%C Convolution of the Catalan sequence 1,1,2,5,14,42,... (A000108) and the Fine sequence 1,0,1,2,6,18,... (A000957). a(n) = A127541(n,0). - _Emeric Deutsch_, Mar 02 2007
%C The Catalan transform of A008619. - _R. J. Mathar_, Nov 06 2008
%C Hankel transform is F(2n+1). - _Paul Barry_, Dec 01 2008
%C Starting with offset 2 = iterates of M * [1,1,0,0,0,...] where M = a tridiagonal matrix with [0,2,2,2,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - _Gary W. Adamson_, Jan 09 2009
%C Equals INVERT transform of A032357. - _Gary W. Adamson_, Apr 10 2009
%C a(n) is the number of Dyck paths of semilength n+1 that have equal length inclines incident with the first return to ground level. For example, for UUDDUUDDUD these inclines are DD and UU (steps 3 through 6), and a(3)=3 counts UDUDUUDD, UDUDUDUD, UUDDUUDD. - _David Callan_, Aug 23 2011
%C a(n) is the number of imprimitive Dyck paths of semilength n+1 for which the heights of the first and the last peaks coincide, this gives the connection to A193215. - _Volodymyr Mazorchuk_, Aug 27 2011
%C a(n) is the number of parking functions of size n-1 avoiding the patterns 123 and 132. - _Lara Pudwell_, Apr 10 2023
%C a(n) is the number of Dyck paths of semilength n that contain no UDUs at ground level. For example, a(3) = 3 counts UUUDDD, UUDUDD, UUDDUD. - _David Callan_, Feb 02 2024
%D Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - _N. J. A. Sloane_, Jun 05 2012
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000958/b000958.txt">Table of n, a(n) for n = 1..1000</a> (first 200 terms from T. D. Noe)
%H Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%H Paul Barry, <a href="https://arxiv.org/abs/2307.00098">Moment sequences, transformations, and Spidernet graphs</a>, arXiv:2307.00098 [math.CO], 2023.
%H Dennis E. Davenport, Louis W. Shapiro, and Leon C. Woodson, <a href="http://math.colgate.edu/~integers/u8/u8.Abstract.html">A bijection between the triangulations of convex polygons and ordered trees</a>, Integers (2020) Vol. 20, Article #A8.
%H E. Deutsch and L. Shapiro, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H Sergio Falcon, <a href="http://www.mathnet.or.kr/mathnet/thesis_file/CKMS-28-4-827-832.pdf">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
%H T. Fine, <a href="http://dx.doi.org/10.1016/S0019-9958(70)90177-4">Extrapolation when very little is known about the source</a>, Information and Control 16 (1970), 331-359.
%H D. G. Rogers, <a href="http://dx.doi.org/10.1016/0097-3165(77)90082-6">Similarity relations on finite ordered sets</a>, J. Combin. Theory, A 23 (1977), 88-98. Erratum, loc. cit., 25 (1978), 95-96.
%H Yidong Sun, <a href="http://dx.doi.org/10.1016/j.disc.2004.07.002">The statistic "number of udu's" in Dyck paths</a>, Discrete Math., 287 (2004), 177-186. See Table 2.
%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = A000957(n) + A000957(n+1).
%F G.f.: (1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2)). - _Paul Barry_, Jan 26 2007
%F G.f.: z*C/(1-z^2*C^2), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. - _Emeric Deutsch_, Mar 02 2007
%F a(n+1) = Sum_{k=0..floor(n/2)} A039599(n-k,k). - _Philippe Deléham_, Mar 13 2007
%F a(n) = (-1/2)^n*(-2 - 5*Sum_{k=1..n-1} (-8)^k*Gamma(1/2+k)*(4/5+k)/(sqrt(Pi)*Gamma(k+3))). - Mark van Hoeij, Nov 11 2009
%F a(n) + a(n+1) = A135339(n+1). - _Philippe Deléham_, Dec 02 2009
%F From _Gary W. Adamson_, Jul 14 2011: (Start)
%F a(n) = sum of top row terms in M^(n-1), where M = the following infinite square production matrix:
%F 0, 1, 0, 0, 0, 0, ...
%F 1, 1, 1, 0, 0, 0, ...
%F 1, 1, 1, 1, 0, 0, ...
%F 1, 1, 1, 1, 1, 0, ...
%F 1, 1, 1, 1, 1, 1, ...
%F ... (End)
%F D-finite with recurrence 2*(n+1)*a(n) + (-5*n+3)*a(n-1) + (-11*n+21)*a(n-2) + 2 *(-2*n+5)*a(n-3) = 0. - _R. J. Mathar_, Dec 03 2012
%F a(n) ~ 5*4^n/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 09 2013
%F a(n) = Catalan(n-1)*h(n-1) for n>=2 where h(n) = hypergeom([1,3/2,-n/2,(1-n)/2],[1/2,-n,-n+1/2], 1). - _Peter Luschny_, Apr 25 2016
%p g:=(1-x-(1+x)*sqrt(1-4*x))/2/x/(x+2): gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=1..26); # _Emeric Deutsch_, Mar 02 2007
%p A958 := n -> add(binomial(2*n-2*k-2, n-1)*(2*k+1)/n, k=0..floor((n-1)/2)): seq(A958(n), n=1..28); # _Johannes W. Meijer_, Jul 26 2013
%p A000958List := proc(m) local A, P, n; A := [1,1]; P := [1,1];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
%p A := [op(A), P[-1]] od; A end: A000958List(28); # _Peter Luschny_, Mar 26 2022
%p # next Maple program:
%p b:= proc(n) option remember; `if`(n<3, n*(2-n),
%p ((7*n-12)*b(n-1)+(4*n-6)*b(n-2))/(2*n))
%p end:
%p a:= n-> b(n)+b(n+1):
%p seq(a(n), n=1..32); # _Alois P. Heinz_, Apr 26 2023
%t nn = 30; Rest[CoefficientList[Series[(1-x-(1+x)*Sqrt[1-4*x])/(2*x*(x+2)), {x, 0, nn}], x]] (* _T. D. Noe_, May 09 2012 *)
%o (Python)
%o from itertools import accumulate
%o def A000958_list(size):
%o if size < 1: return []
%o L, accu = [], [1]
%o for n in range(size-1):
%o accu = list(accumulate(accu+[-accu[-1]]))
%o L.append(accu[n])
%o return L
%o print(A000958_list(29)) # _Peter Luschny_, Apr 25 2016
%o (Python)
%o from itertools import count, islice
%o def A000958_gen(): # generator of terms
%o yield 1
%o a, c = 0, 1
%o for n in count(1):
%o yield (c:=c*((n<<2)+2)//(n+2))+a>>1
%o a = c-a>>1
%o A000958_list = list(islice(A000958_gen(),20)) # _Chai Wah Wu_, Apr 26 2023
%o (PARI) my(x='x+O('x^30)); Vec((1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2))) \\ _G. C. Greubel_, Feb 27 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x-(1+x)*Sqrt(1-4*x))/(2*x*(x+2)) )); // _G. C. Greubel_, Feb 27 2019
%o (Sage) a=((1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 27 2019
%Y First column of A065602, A098747 and A362563. Row sums of A362563.
%Y Cf. A127541, A127539, A000108, A000957, A032357.
%Y Partial differences give A118973 (for n>=1).
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
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