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%I M1220
%S 1,2,4,10,26,76,232,764,2619,9486,35596,139392,562848,2352064,
%T 10092160,44546320,201158620,930213752,4387327088,21115314916,
%U 103386386516,515097746072,2605341147472,13378787264584,69622529312665,367161088308490,1959294979429380
%N Number of Young tableaux of height <= 8.
%C Also the number of n-length words w over 8-ary alphabet {a1,a2,...,a8} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a8), where #(z,x) counts the number of letters x in word z. - _Alois P. Heinz_, May 30 2012
%D F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A007580/b007580.txt">Table of n, a(n) for n = 1..400</a>
%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%p h:= proc(l) local n; n:=nops(l);
%p add(i, i=l)! /mul (mul (1+l[i]-j +add (`if`(l[k]>=j, 1, 0),
%p k=i+1..n), j=1..l[i]), i=1..n)
%p end:
%p g:= proc(n, i, l) option remember;
%p `if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
%p g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p end:
%p a:= n-> g(n, 8, []):
%p seq (a(n), n=1..30); # _Alois P. Heinz_, Apr 10 2012
%p # second Maple program
%p a:= proc(n) option remember;
%p `if`(n<4, [1, 1, 2, 4][n+1],
%p ((40*n^3+1084*n^2+8684*n+18480)*a(n-1)
%p +16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)
%p -1024*(n-1)*(n-2)*(n+6)*a(n-3)
%p -1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /
%p ((n+7)*(n+12)*(n+15)*(n+16)))
%p end:
%p seq (a(n), n=1..30); # _Alois P. Heinz_, Oct 12 2012
%Y Column k=8 of A182172. - _Alois P. Heinz_, May 30 2012
%K nonn
%O 1,2
%A _Simon Plouffe_
%E More terms from _Alois P. Heinz_, Apr 10 2012
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