OFFSET
5,2
COMMENTS
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
Number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014
REFERENCES
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).
LINKS
T. D. Noe, Table of n, a(n) for n = 5..100
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
Arthur 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.
Petr Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation, Vol. 20, No. 5-6 (1995), 595-601.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
Ronald C. Read, On general dissections of a polygon, Preprint (1974).
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76 (1996), 175-177.
Hua Xin, Lattice points of flow polytopes related to caracol graphs, AIMS Elect. Res. Archive 33(10) (2025) 6141-6175. See p. 6159, Table 4.
FORMULA
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
a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017
From Amiram Eldar, Oct 24 2025: (Start)
Sum_{n>=5} 1/a(n) = 13 - 16*Pi/(3*sqrt(3)) - 2*Pi^2/9.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*log(phi)^2 + 56*log(phi)/sqrt(5) - 13, where phi is the golden ratio (A001622). (End)
MATHEMATICA
Table[(Binomial[n-3, 2]Binomial[2n-6, n-5])/(n-4), {n, 5, 30}] (* Harvey P. Dale, Nov 06 2011 *)
PROG
(PARI) a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);
for(n=5, 30, print1(a(n), ", ")) \\ Indranil Ghosh, Apr 11 2017
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved
