A215541 a(n) = binomial(5*n,n)*(3*n+1)/(4*n+1).

1, 4, 35, 350, 3705, 40480, 451269, 5101360, 58261125, 670609940, 7766844470, 90404916420, 1056658719675, 12393263030400, 145787921878840, 1719353829326880, 20322351313767965, 240674861588534100, 2855214354095519625, 33924757188414045330, 403641797464597415570

Offset

0,2

Comments

Number of standard Young tableaux of shape [4n,n]. Also the number of binary words with 4n 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) = 4 words are: 10111, 11011, 11101, 11110.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..285

Wikipedia, Young tableau.

Formula

a(n) = C(5*n,n)*(3*n+1)/(4*n+1).

a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(3*n+1). - Ilya Gutkovskiy, Nov 01 2017

Recurrence: 8*n*(2*n - 1)*(3*n - 2)*(4*n - 1)*(4*n + 1)*a(n) = 5*(3*n + 1)*(5*n - 4)*(5*n - 3)*(5*n - 2)*(5*n - 1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018

a(n) ~ 3 * 5^(5*n+1/2) / (2^(8*n+7/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 5^5*x/2^8) + 3*x*hypergeom([6/5, 7/5, 8/5, 9/5], [3/2, 7/4, 9/4], 5^5*x/2^8). - Stefano Spezia, Sep 11 2025

Maple

a:= n-> binomial(5*n, n)*(3*n+1)/(4*n+1):
seq(a(n), n=0..25);

Mathematica

a[n_] := Binomial[5*n, n]*(3*n+1)/(4*n+1); Array[a, 21, 0] (* Amiram Eldar, Aug 29 2025 *)

Crossrefs

Column k=4 of A214776.

Sequence in context: A389936 A292190 A026304 * A390572 A386979 A104456

Adjacent sequences: A215538 A215539 A215540 * A215542 A215543 A215544

Keyword

nonn

Author

Alois P. Heinz, Aug 15 2012

Status

approved