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%I #55 Mar 07 2023 11:19:50
%S 0,0,1,1,1,2,4,8,17,37,82,185,423,978,2283,5373,12735,30372,72832,
%T 175502,424748,1032004,2516347,6155441,15101701,37150472,91618049,
%U 226460893,560954047,1392251012,3461824644,8622571758,21511212261,53745962199
%N Number of elevated peakless Motzkin paths.
%C Essentially the same as A004148: a(0)=a(1)=0 and a(n) = A004148(n-2) for n>=2.
%D A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.
%D A. Panayotopoulos and P. Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.
%H Muniru A Asiru, <a href="/A203019/b203019.txt">Table of n, a(n) for n = 0..300</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H I. Jensen, <a href="http://arxiv.org/abs/cond-mat/9910313">Enumeration of plane meanders</a>, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
%H S. K. Lando and A. K. Zvonkin, <a href="http://dx.doi.org/10.1016/0304-3975(93)90316-L">Plane and projective meanders</a>, Theoretical Computer Science Vol. 117 (1993) p. 232.
%H A. Panayotopoulos and P. Tsikouras, <a href="https://doi.org/10.4000/msh.2808">The multimatching property of nested sets</a>, Math. & Sci. Hum. 149 (2000), 23-30.
%H A. Panayotopoulos and P. Tsikouras, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Panayotopoulos/panayo4.html">Meanders and Motzkin Words</a>, J. Integer Seqs., Vol. 7, 2004.
%H A. Panayotopoulos and P. Vlamos, <a href="http://dx.doi.org/10.1007/978-3-642-33412-2_49">Cutting Degree of Meanders</a>, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012
%F G.f.: x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / ...)))))). - _Michael Somos_, May 12 2012
%F G.f. A(x) =: y satisfies y / x = x + y / (1 - y). - _Michael Somos_, Jan 31 2014
%F G.f. A(x) =: y satisfies y = x^2 + (x - x^2)*y + y*y. - _Michael Somos_, Jan 31 2014
%F Given g.f. A(x), then B(x) = A(x)/x satisfies B(-B(-x)) = x. - _Michael Somos_, Jan 31 2014
%F a(n) = Sum_{m=0..(n-1)/2}((binomial(2*m+1,m)*Sum_{k=0..n-2*m-2}(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k)))/(2*m+1)). - _Vladimir Kruchinin_, Mar 12 2016
%F a(n) ~ 5^(1/4) * phi^(2*n - 2) / (2*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 14 2018
%F D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023
%e G.f. = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 17*x^8 + 37*x^9 + ...
%t terms = 34;
%t A[_] = 0; Do[A[x_] = x (x - A[x] / (A[x] - 1)) + O[x]^terms, {terms}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 27 2018, after _Michael Somos_ *)
%t Table[Sum[Binomial[2*m + 1, m]*Sum[(Binomial[k, n - 2*m - k - 2]* Binomial[2*m + k, k]*(-1)^(n - k))/(2*m + 1), {k, 0, n - 2*m - 2}], {m, 0, Floor[(n - 1)/2]}], {n, 0, 50}] (* _G. C. Greubel_, Aug 12 2018 *)
%o (PARI) {a(n) = local(A); A = O(x); for( k=1, ceil(n / 3), A = x^2 / (1 - x / (1 - A))); polcoeff( A, n)} /* _Michael Somos_, May 12 2012 */
%o (Maxima)
%o a(n):=sum((binomial(2*m+1,m)*sum(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k),k,0,n-2*m-2))/(2*m+1),m,0,(n-1)/2); /* _Vladimir Kruchinin_, Mar 12 2016 */
%o (GAP) List([0..40],n->Sum([0..Int((n-1)/2)],m->Binomial(2*m+1,m)*Sum([0..n-2*m-2],k->(Binomial(k,n-2*m-k-2)*Binomial(2*m+k,k)*(-1)^(n-k))/(2*m+1)))); # _Muniru A Asiru_, Aug 13 2018
%K nonn
%O 0,6
%A _Panayotis Vlamos_ and _Antonios Panayotopoulos_, Dec 27 2011
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