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A007297
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Number of connected graphs on n nodes on a circle without crossing edges.
(Formerly M3594)
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14
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1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Reversion of g.f. for squares (ignoring signs).
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REFERENCES
| C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358 (column sums in Table 2).
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math. 204 (1999), 203-229.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486
Index entries for reversions of series
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FORMULA
| REVERT(A000290).
G.f.: (g-z)/z, where g=-1/3+(2/3)*sqrt(1+9z)*sin((1/3)*arcsin((2+27z+54z^2)/2/(1+9*z)^(3/2))); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 02 2002
a(n)=(1/n)*sum{k=0..n, binomial(3n, n-k-1)*binomial(n+k-1, k)}; - Paul Barry (pbarry(AT)wit.ie), May 11 2005
(Maple notation) an := 4^n*(GAMMA((3*n+1)/2)/GAMMA((n+3)/2)/GAMMA(n+1) -GAMMA( 3*n/2+1 )/GAMMA( n/2 +1)/GAMMA(n+2)); - Mark van Hoeij (Florida State Univ.), Aug 27 2005
C := binomial; an := 4^(n+1) * C(3*(n+1)/2, (n+1)/2) / (9*n+3) - 4^n * C(3*n/2, n/2 ) / (n+1); - Mark van Hoeij (Florida State Univ.), Aug 27 2005
-12*(3*n+2)*(3*n+1)*(3*n+8)*a(n)+(72+36*n)*a(n+1)+(3*n+5)*(n+3)*(n+2)*a(n+2) = 0 - Mark van Hoeij (Florida State Univ.), Aug 27 2005
a(n)=(1/n)*sum{k=0..n, C(3n, k)C(2n-k-2, n-1)}; - Paul Barry (pbarry(AT)wit.ie), Sep 27 2005
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MAPLE
| add(binomial(3*n - 3, n + j)*binomial(j - 1, j - n + 1), j = n - 1 .. 2*n - 3)/(n - 1);
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MATHEMATICA
| CoefficientList[ InverseSeries[ Series[(x-x^2)/(1+x)^3, {x, 0, 20}], x], x] // Rest (* From Jean-François Alcover, May 19 2011, after PARI prog. *)
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-x^2)/(1+x)^3+O(x^(n+2))), n+1)) (from R. Stephan)
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CROSSREFS
| Sequence in context: A193113 A192730 A055723 * A198916 A182969 A111547
Adjacent sequences: A007294 A007295 A007296 * A007298 A007299 A007300
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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EXTENSIONS
| Better description from Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Apr 20 2000
More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 21 2000
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