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A203019 Number of elevated peakless Motzkin paths. 2
0, 0, 1, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Essentially the same as A004148: a(0)=a(1)=0 and a(n) = A004148(n-2) for n>=2.

REFERENCES

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

A. Panayotopoulos and P. Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..300

I. Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.

S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993) p. 232.

A. Panayotopoulos and P. Tsikouras, The multimatching property of nested sets, Math. & Sci. Hum. 149 (2000), 23-30.

A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.

A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, 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

FORMULA

G.f.: x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / ...)))))). - Michael Somos, May 12 2012

G.f. A(x) =: y satisfies y / x = x + y / (1 - y). - Michael Somos, Jan 31 2014

G.f. A(x) =: y satisfies y = x^2 + (x - x^2)*y + y*y. - Michael Somos, Jan 31 2014

Given g.f. A(x), then B(x) = A(x)/x satisfies B(-B(-x)) = x. - Michael Somos, Jan 31 2014

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

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

EXAMPLE

G.f. = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 17*x^8 + 37*x^9 + ...

MATHEMATICA

terms = 34;

A[_] = 0; Do[A[x_] = x (x - A[x] / (A[x] - 1)) + O[x]^terms, {terms}];

CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Jul 27 2018, after Michael Somos *)

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 *)

PROG

(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 */

(Maxima)

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 */

(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

CROSSREFS

Sequence in context: A199409 A025241 A292461 * A004148 A292460 A085022

Adjacent sequences:  A203016 A203017 A203018 * A203020 A203021 A203022

KEYWORD

nonn,changed

AUTHOR

Panayotis Vlamos and Antonios Panayotopoulos, Dec 27 2011

STATUS

approved

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Last modified August 15 04:21 EDT 2018. Contains 313756 sequences. (Running on oeis4.)