OFFSET
0,6
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
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
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
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
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-François 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
KEYWORD
nonn
AUTHOR
Panayotis Vlamos and Antonios Panayotopoulos, Dec 27 2011
STATUS
approved