OFFSET
0,2
COMMENTS
Number of standard tableaux of shape (n+1,n+1,1,1,1,1,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
From Oliver Pechenik, May 02 2014: (Start)
Number of increasing tableaux of shape (n+7,n+7) with largest entry 2*n+8. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers.
a(n) = number of noncrossing partitions of 2*n+8 into n+1 blocks all of size at least 2. (End)
LINKS
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
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.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76 (1996), 175-177.
FORMULA
a(n) = binomial(n+6, 6)*binomial(2*n+8, n)/(n+1).
From Amiram Eldar, Nov 04 2025: (Start)
a(n) ~ 2^(2*n+4) * n^(9/2) / (45 * sqrt(Pi)).
Sum_{n>=0} 1/a(n) = 71614/5 - 2106*sqrt(3)*Pi - 290*Pi^2.
Sum_{n>=0} (-1)^n/a(n) = -7354/5 + 2220*sqrt(5)*log(phi) - 3960*log(phi)^2, where phi is the golden ratio (A001622). (End)
MATHEMATICA
a[n_] := Binomial[n+6, 6] * Binomial[2*n+8, n]/(n+1); Array[a, 20, 0] (* Amiram Eldar, Nov 04 2025 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Amiram Eldar, Nov 04 2025
STATUS
approved
