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A002055
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Number of diagonal dissections of a convex n-gon into n-4 regions.
(Formerly M4639 N1982)
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9
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1, 9, 56, 300, 1485, 7007, 32032, 143208, 629850, 2735810, 11767536, 50220040, 212952285, 898198875, 3771484800, 15775723920, 65770848990, 273420862110, 1133802618000, 4691140763400, 19371432850770, 79850555673174
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OFFSET
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5,2
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COMMENTS
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Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014
a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014
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REFERENCES
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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).
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LINKS
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A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
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FORMULA
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a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).
With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - Paul Barry, Aug 25 2007
G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - R. J. Mathar, Nov 17 2011
D-finite with recurrence: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
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MATHEMATICA
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Table[(Binomial[n-3, 2]Binomial[2n-6, n-5])/(n-4), {n, 5, 30}] (* Harvey P. Dale, Nov 06 2011 *)
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PROG
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(PARI) a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);
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CROSSREFS
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a(n) = f(n,n+1) where f is given in A034261.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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