A215541 - OEIS

%I #18 Feb 03 2018 04:23:10

%S 1,4,35,350,3705,40480,451269,5101360,58261125,670609940,7766844470,

%T 90404916420,1056658719675,12393263030400,145787921878840,

%U 1719353829326880,20322351313767965,240674861588534100,2855214354095519625,33924757188414045330,403641797464597415570

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

%C 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.

%H Alois P. Heinz, <a href="/A215541/b215541.txt">Table of n, a(n) for n = 0..285</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

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

%F a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(3*n+1). - _Ilya Gutkovskiy_, Nov 01 2017

%F 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

%p a:= n-> binomial(5*n,n)*(3*n+1)/(4*n+1):

%p seq(a(n), n=0..25);

%Y Column k=4 of A214776.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Aug 15 2012