A006664 Number of irreducible systems of meanders.

1, 1, 2, 8, 46, 322, 2546, 21870, 199494, 1904624, 18846714, 191955370, 2002141126, 21303422480, 230553207346, 2531848587534, 28159614749270, 316713536035464, 3597509926531778, 41225699113145888, 476180721050626814, 5539597373695447322, 64863295574835126394, 763984568163192551672, 9047263176444565467566

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice
• changed

References

• V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
• S. K. Lando and A. K. Zvonkin, "Plane and projective meanders", Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303.
• S. K. Lando and A. K. Zvonkin, "Meanders", Selecta Mathematica Sovietica Vol. 11, Number 2, pp. 117-144, 1992.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.

Programs

Mathematica

terms = 25;
S[x_] = Sum[CatalanNumber[k]^2 x^k, {k, 0, 2 terms}];
inv = InverseSeries[x S[x^2] + O[x]^(2 terms), x] // Normal;
(S[x^2] /. x -> inv) + O[x]^(2 terms) // CoefficientList[#, x]& // DeleteCases[#, 0]& (* Jean-François Alcover, Sep 04 2018 *)
terms = 25;
S = Sum[CatalanNumber[k]^2 x^k, {k, 0, terms}] + O[x]^terms;
ComposeSeries[S,InverseSeries[x S^2]] // CoefficientList[#, x] &
(* Mamuka Jibladze, Jun 04 2025 *)

Formula

A(x^2) = S(x^2)#inv(x*S(x^2)) where # is functional composition, S(x) is g.f. of A001246 (squares of Catalan numbers) and inv(.) is functional inverse. A(x) consists of even-numbered terms of A(x^2), odd terms of which are 0. - Doug McIlroy (doug(AT)cs.dartmouth.edu), Mar 22 2006

A(x) = S(x)#inv(x*S(x)^2) where # is functional composition, S(x) is g.f. of A001246 (squares of Catalan numbers) and inv(.) is functional inverse. - Mamuka Jibladze, Jun 04 2025

A(x) satisfies ODE A^2*(A^2 - A - 4*x) - x*A*(3*A^2 - 6*A + 8*x)*A' + 4*x^2*(A^2 - 3*A + 4*x)*(A')^2 - 2*x^3*(A - 4)*(A')^3 + x^2*A*(A^2 - 16*x)*A'' = 0 (derived from the functional equation using S(x^2) = (2F_1(-1/2,-1/2,1,16x^2)-1) / (4x^2)). - _Mamuka Jibladze, Aug 25 2025

Extensions

More terms from Doug McIlroy (doug(AT)cs.dartmouth.edu), Mar 22 2006