A126596 a(n) = binomial(4*n,n)*(2*n+1)/(3*n+1).

1, 3, 20, 154, 1260, 10659, 92092, 807300, 7152444, 63882940, 574221648, 5188082354, 47073334100, 428634152730, 3914819231400, 35848190542920, 329007937216860, 3025582795190340, 27872496751392496, 257172019222240200, 2376196095585231920, 21983235825545286435

Offset: 0

Author

Philippe Deléham, Mar 13 2007

Keywords

• nonn
• easy
• changed

Comments

Number of standard Young tableaux of shape [3n,n]. Also the number of binary words with 3n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 3 words are: 1011, 1101, 1110. - Alois P. Heinz, Aug 15 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Column k=3 of A214776.

Programs

Haskell

a126596 n = a005810 n * a005408 n `div` a016777 n
-- Reinhard Zumkeller, Mar 04 2012

Magma

[Binomial(4*n,n)*(2*n+1)/(3*n+1): n in [0..20]]; // Vincenzo Librandi, Nov 18 2011

Maple

seq((2*n+1)*binomial(4*n,n)/(3*n+1),n=0..22); # Emeric Deutsch, Mar 27 2007

Mathematica

Table[(Binomial[4n,n](2n+1))/(3n+1),{n,0,30}] (* Harvey P. Dale, Feb 06 2016 *)

Formula

a(n) = A039599(2*n,n).

a(n) = (2*n+1)*A002293(n). - Mark van Hoeij, Nov 17 2011

a(n) = A208983(2*n+1). - Reinhard Zumkeller, Mar 04 2012

a(n) = A005810(n) * A005408(n) / A016777(n). - Reinhard Zumkeller, Mar 04 2012

a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(2*n+1). - Ilya Gutkovskiy, Nov 01 2017

Recurrence: 3*n*(3*n-1)*(3*n+1)*a(n) = 8*(2*n+1)*(4*n-3)*(4*n-1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018

a(n) ~ 2^(8*n+3/2) / (3^(3*n+3/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

Extensions

More terms from Emeric Deutsch, Mar 27 2007