This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006664 M1871 #46 Aug 28 2025 21:58:37
%S A006664 1,1,2,8,46,322,2546,21870,199494,1904624,18846714,191955370,
%T A006664 2002141126,21303422480,230553207346,2531848587534,28159614749270,
%U A006664 316713536035464,3597509926531778,41225699113145888,476180721050626814,5539597373695447322,64863295574835126394,763984568163192551672,9047263176444565467566
%N A006664 Number of irreducible systems of meanders.
%D A006664 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.
%D A006664 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.
%D A006664 S. K. Lando and A. K. Zvonkin, "Meanders", Selecta Mathematica Sovietica Vol. 11, Number 2, pp. 117-144, 1992.
%D A006664 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006664 Motohisa Fukuda, Ion Nechita, <a href="https://arxiv.org/abs/1609.02756">Enumerating meandric systems with large number of components</a>, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
%H A006664 S. K. Lando and A. K. Zvonkin, <a href="/A005316/a005316_1.pdf">Plane and projective meanders</a>, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
%H A006664 S. K. Lando and A. K. Zvonkin, <a href="https://doi.org/10.1016/0304-3975(93)90316-L">Plane and projective meanders</a>, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
%F A006664 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
%F A006664 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
%F A006664 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
%t A006664 terms = 25;
%t A006664 S[x_] = Sum[CatalanNumber[k]^2 x^k, {k, 0, 2 terms}];
%t A006664 inv = InverseSeries[x S[x^2] + O[x]^(2 terms), x] // Normal;
%t A006664 (S[x^2] /. x -> inv) + O[x]^(2 terms) // CoefficientList[#, x]& // DeleteCases[#, 0]& (* _Jean-François Alcover_, Sep 04 2018 *)
%t A006664 terms = 25;
%t A006664 S = Sum[CatalanNumber[k]^2 x^k, {k, 0, terms}] + O[x]^terms;
%t A006664 ComposeSeries[S,InverseSeries[x S^2]] // CoefficientList[#, x] &
%t A006664 (* _Mamuka Jibladze_, Jun 04 2025 *)
%K A006664 nonn,nice,changed
%O A006664 0,3
%A A006664 _N. J. A. Sloane_, _Simon Plouffe_
%E A006664 More terms from Doug McIlroy (doug(AT)cs.dartmouth.edu), Mar 22 2006