A215543 Number of standard Young tableaux of shape [3n,3].

0, 5, 48, 154, 350, 663, 1120, 1748, 2574, 3625, 4928, 6510, 8398, 10619, 13200, 16168, 19550, 23373, 27664, 32450, 37758, 43615, 50048, 57084, 64750, 73073, 82080, 91798, 102254, 113475, 125488, 138320, 151998, 166549, 182000, 198378, 215710, 234023, 253344

Offset: 0

Author

Alois P. Heinz, Aug 15 2012

Keywords

• nonn
• easy
• changed

Comments

Also the number of binary words with 3n 1's and 3 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 5 words are: 101010, 101100, 110010, 110100, 111000.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Young tableau.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Row n=3 of A214776.

Programs

Maple

a:= n-> max(0, (3*n-2)*(3*n+2)*(n+1)/2):
seq(a(n), n=0..40);

Mathematica

LinearRecurrence[{4,-6,4,-1},{0,5,48,154,350},50] (* Harvey P. Dale, Dec 16 2017 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;5;48;154])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

Formula

G.f.: (2*x^3-8*x^2+28*x+5)*x/(x-1)^4.

a(n) = (3*n-2)*(3*n+2)*(n+1)/2 for n>0, a(0) = 0.

From Amiram Eldar, Aug 29 2025: (Start)

Sum_{n>=1} 1/a(n) = 7/20 + sqrt(3)*Pi/10 - 3*log(3)/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/5 - 4*log(2)/5 - 7/20. (End)