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A002740
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Tree-rooted planar maps.
(Formerly M2078 N0821)
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5
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0, 0, 0, 2, 15, 84, 420, 1980, 9009, 40040, 175032, 755820, 3233230, 13728792, 57946200, 243374040, 1017958725, 4242920400, 17631691440, 73078721100, 302202005490, 1247182879800, 5137916074200, 21132472200840
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=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*A129175(n,k),k=0..n(n-1)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2007
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REFERENCES
| M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106.
J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, A, 40, 248-264, 1985.
J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 97.
M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Walsh, T. R. S.; Lehman, A. B.; Counting rooted maps by genus. III: Nonseparable maps. J. Combinatorial Theory Ser. B 18 (1975), 222-259.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals
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FORMULA
| G.f.: (1/2)*(1-(1-6*t+6*t^2)/(1-4*t)^(3/2)).
a(n+3)=(2*(n+2))!/(2*n!*(n+3)!), from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
a(n+2)=sum(k=0, n, k*binomial(k+n, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 25 2003
a(n)=sum(sum(binomial(2*n,n)/(n+1)/2, j=1..n),k=2..n), n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
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MAPLE
| 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 (zerinvarylajos(AT)yahoo.com), Mar 13 2007
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 (zerinvarylajos(AT)yahoo.com), May 09 2007
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MATHEMATICA
| a[n_] := (n-1)(n-2)Binomial[2(n-1), n-1]/(2n); a[0] = 0; Table[a[n], {n, 0, 23}] (* From Jean-François Alcover, Nov 16 2011 *)
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PROG
| (Mupad) combinat::catalan(n) *binomial(n, 2) $ n = 0..22 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007
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CROSSREFS
| Cf. A129175.
Sequence in context: A036239 A109725 A057152 * A178750 A108475 A098624
Adjacent sequences: A002737 A002738 A002739 * A002741 A002742 A002743
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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