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A007580 Number of Young tableaux of height <= 8.

%I A007580 M1220 #44 Dec 20 2017 23:28:58
%S A007580 1,1,2,4,10,26,76,232,764,2619,9486,35596,139392,562848,2352064,
%T A007580 10092160,44546320,201158620,930213752,4387327088,21115314916,
%U A007580 103386386516,515097746072,2605341147472,13378787264584,69622529312665,367161088308490,1959294979429380
%N A007580 Number of Young tableaux of height <= 8.
%C A007580 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 letters x in word z. - _Alois P. Heinz_, May 30 2012
%D A007580 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007580 Alois P. Heinz, <a href="/A007580/b007580.txt">Table of n, a(n) for n = 0..1000</a>
%H A007580 F. Bergeron, L. Favreau and D. Krob, <a href="/A007578/a007578.pdf">Conjectures on the enumeration of tableaux of bounded height</a>, Preprint. (Annotated scanned copy)
%H A007580 F. Bergeron, L. Favreau and D. Krob, <a href="http://dx.doi.org/10.1016/0012-365X(94)00148-C">Conjectures on the enumeration of tableaux of bounded height</a>, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
%H A007580 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A007580 a(n) ~ 135/16 * 8^(n+14)/(Pi^2*n^14). - _Vaclav Kotesovec_, Sep 11 2013
%p A007580 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
%p A007580       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A007580     end:
%p A007580 g:= proc(n, i, l) option remember;
%p A007580       `if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
%p A007580         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A007580     end:
%p A007580 a:= n-> g(n, 8, []):
%p A007580 seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 10 2012
%p A007580 # second Maple program:
%p A007580 a:= proc(n) option remember;
%p A007580       `if`(n<4, [1, 1, 2, 4][n+1],
%p A007580        ((40*n^3+1084*n^2+8684*n+18480)*a(n-1)
%p A007580        +16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)
%p A007580        -1024*(n-1)*(n-2)*(n+6)*a(n-3)
%p A007580        -1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /
%p A007580        ((n+7)*(n+12)*(n+15)*(n+16)))
%p A007580     end:
%p A007580 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 12 2012
%t A007580 RecurrenceTable[{1024 (-3+n) (-2+n) (-1+n) (4+n) a[-4+n]+1024 (-2+n) (-1+n) (6+n) a[-3+n]-16 (-1+n) (600+610 n+107 n^2+5 n^3) a[-2+n]-4 (4620+2171 n+271 n^2+10 n^3) a[-1+n]+(7+n) (12+n) (15+n) (16+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10}, a, {n, 20}] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%Y A007580 Column k=8 of A182172. - _Alois P. Heinz_, May 30 2012
%K A007580 nonn
%O A007580 0,3
%A A007580 _Simon Plouffe_
%E A007580 More terms from _Alois P. Heinz_, Apr 10 2012