cp's OEIS Frontend

A002740 Number of tree-rooted bridgeless planar maps with two vertices and n faces.

%I A002740 M2078 N0821 #92 Feb 12 2023 07:19:31
%S A002740 0,0,0,2,15,84,420,1980,9009,40040,175032,755820,3233230,13728792,
%T A002740 57946200,243374040,1017958725,4242920400,17631691440,73078721100,
%U A002740 302202005490,1247182879800,5137916074200,21132472200840,86794082253450,356013544661424,1458583920435600,5969389748449400
%N A002740 Number of tree-rooted bridgeless planar maps with two vertices and n faces.
%C A002740 a(n) is the sum of the major indices of all Dyck words of length 2n-2. The major index of a Dyck word is the sum of the positions of those 1's that are followed by a 0. Example: a(4)=15 because the Dyck words of length 6 are 010101, 010011, 001101, 001011 and 000111 having major indices 6,2,4,3 and 0, respectively. a(n) = Sum_{k=0..n(n-1)} k*A129175(n,k). - _Emeric Deutsch_, Apr 20 2007
%D A002740 J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 97.
%D A002740 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002740 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002740 Vincenzo Librandi, <a href="/A002740/b002740.txt">Table of n, a(n) for n = 0..200</a>
%H A002740 Mireille Bousquet-Mélou, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00109-X">New enumerative results on two-dimensional directed animals</a>, Discr. Math., Vol. 180, No. 1-3 (1998), pp. 73-106. See Cor. 6.6.
%H A002740 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6807">Enumeration of saturated chains in Dyck lattices</a>, arXiv preprint arXiv:1203.6807 [math.CO], 2012.
%H A002740 J. Fürlinger and J. Hofbauer, <a href="http://dx.doi.org/10.1016/0097-3165(85)90089-5">q-Catalan numbers</a>, J. Comb. Theory, Series A, Vol. 40, No. 2 (1985), pp. 248-264.
%H A002740 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H A002740 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A002740 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933. (Annotated scans of some selected pages)
%H A002740 Mark Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/f7/f7.Abstract.html">Parity theorems for statistics on permutations and Catalan words</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5 (2005), Paper A07.
%H A002740 T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259. See Table VIIIb.
%F A002740 G.f.: (1/2)*(1-(1 - 6*t + 6*t^2)/(1-4*t)^(3/2)).
%F A002740 a(n+3) = (2*(n+2))!/(2*n!*(n+3)!). - _Wolfdieter Lang_
%F A002740 a(n+2) = Sum_{k=0..n} k*binomial(k+n, k). - _Benoit Cloitre_, Oct 25 2003
%F A002740 a(n) = Sum_{k=2..n} Sum_{j=1..n} binomial(2*n,n)/(2*(n+1)), n >= 0. - _Zerinvary Lajos_, May 09 2007
%F A002740 a(n) = (n-2)*binomial(2n-2, n-2)/2 + 0^n. - _Wesley Ivan Hurt_, Sep 24 2014
%F A002740 E.g.f.: (1 + exp(2*x) * ((2*x - 1) * BesselI(0,2*x) - x * BesselI(1,2*x))) / 2. - _Ilya Gutkovskiy_, Nov 03 2021
%F A002740 From _Amiram Eldar_, Mar 22 2022: (Start)
%F A002740 Sum_{n>=3} 1/a(n) = 3 - 4*Pi/(3*sqrt(3)).
%F A002740 Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(phi)/sqrt(5) - 3, where phi is the golden ratio (A001622). (End)
%p A002740 with(combinat):for n from 0 to 22 do printf(`%d, `,n*sum(binomial(2*n, n)/(n+1)/2, k=2..n)) od: # _Zerinvary Lajos_, Mar 13 2007
%p A002740 a:=n->sum(sum(binomial(2*n,n)/(n+1)/2, j=1..n),k=2..n): seq(a(n), n=0..25); # _Zerinvary Lajos_, May 09 2007
%p A002740 A002740:=n->(n-2)*binomial(2*n-2,n-2)/2+0^n: seq(A002740(n), n=0..30); # _Wesley Ivan Hurt_, Sep 24 2014
%t A002740 a[n_] := (n-1)(n-2)Binomial[2(n-1), n-1]/(2n); a[0] = 0; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Nov 16 2011 *)
%o A002740 (MuPAD) combinat::catalan(n) *binomial(n,2) $ n = 0..22 // _Zerinvary Lajos_, Feb 15 2007
%o A002740 (PARI) a(n)=if(n<3,0,(2*(n-1))!/(2*n!*(n-3)!)); /* _Joerg Arndt_, Sep 28 2012 */
%o A002740 (Magma) [(n-2)*Binomial(2*n-2,n-2)/2 : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 24 2014
%Y A002740 Cf. A129175.
%Y A002740 A diagonal of A253180.
%K A002740 nonn,easy
%O A002740 0,4
%A A002740 _N. J. A. Sloane_
%E A002740 Name clarified by _Noam Zeilberger_, Aug 18 2017