%I #12 Oct 21 2022 21:38:59

%S 0,42,1638,10659,40480,115101,272272,566618,1072764,1888460,3137706,

%T 4973877,7582848,11186119,16043940,22458436,30776732,41394078,

%U 54756974,71366295,91780416,116618337,146562808,182363454,224839900,274884896,333467442,401635913

%N Number of standard Young tableaux of shape [5n,5].

%C Also the number of binary words with 5n 1's and 5 0's such that for every prefix the number of 1's is >= the number of 0's.

%H Alois P. Heinz, <a href="/A215545/b215545.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: (4*x^5-24*x^4+256*x^3+1461*x^2+1386*x+42)*x/(x-1)^6.

%F a(n) = (5*n-4)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)/24 for n>0, a(0) = 0.

%p a:= n-> max(0, (5*n-4)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)/24):

%p seq(a(n), n=0..40);

%Y Row n=5 of A214776.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Aug 15 2012