A215544 Number of standard Young tableaux of shape [4n,4].

0, 14, 275, 1260, 3705, 8602, 17199, 31000, 51765, 81510, 122507, 177284, 248625, 339570, 453415, 593712, 764269, 969150, 1212675, 1499420, 1834217, 2222154, 2668575, 3179080, 3759525, 4416022, 5154939, 5982900, 6906785, 7933730, 9071127, 10326624, 11708125

Offset: 0

Author

Alois P. Heinz, Aug 15 2012

Keywords

• nonn
• easy
• changed

Comments

Also the number of binary words with 4n 1's and 4 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 14 words are: 10101010, 10101100, 10110010, 10110100, 10111000, 11001010, 11001100, 11010010, 11010100, 11011000, 11100010, 11100100, 11101000, 11110000.

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 (5,-10,10,-5,1).

Crossrefs

Row n=4 of A214776.

Programs

Maple

a:= n-> max(0, (4*n+3)*(2*n+1)*(4*n-3)*(n+1)/3):
seq(a(n), n=0..40);

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{0,14,275,1260,3705,8602},40] (* Harvey P. Dale, Jan 25 2024 *)

PARI

a(n)=max((4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3,0) \\ Charles R Greathouse IV, Oct 21 2022

Formula

G.f.: (3*x^4-15*x^3-25*x^2-205*x-14)*x/(x-1)^5.

a(n) = (4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3 for n>0, a(0) = 0.

Sum_{n>=1} 1/a(n) = 31/105 - 17*Pi/35 + 66*log(2)/35. - Amiram Eldar, Aug 29 2025