%I #16 Jan 02 2021 08:01:03

%S 1,1,6,51,588,7990,126060,2242618,44546320,977152266,23500234512,

%T 615372604033,17442275104496,532242021137346,17399782340548920,

%U 606732491690590816,22477989291826848000,881635273413199806994,36493478646922003374096,1589642562747880936613248

%N Number of standard Young tableaux of 2n cells and height <= n.

%C Also the number of standard Young tableaux of 2n cells and <= n columns.

%C Also the number of 2n-length words w over n-ary alphabet {a1,a2,...,an} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,an), where #(z,x) counts the letters x in word z. The a(2) = 6 words of length 4 over alphabet {a,b} are: aaaa, aaab, aaba, abaa, aabb, abab.

%H Vaclav Kotesovec, <a href="/A293128/b293128.txt">Table of n, a(n) for n = 0..41</a> (terms 0..32 from Alois P. Heinz)

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

%F a(n) = A182172(2n,n).

%p h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]

%p <j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):

%p g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), add(

%p g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):

%p a:= n-> g(2*n, n, []):

%p seq(a(n), n=0..15);

%t h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];

%t g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];

%t a[n_] := g[2n, n, {}];

%t a /@ Range[0, 15] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)

%Y Cf. A182172, A267436.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 30 2017