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A007578 Number of Young tableaux of height <= 7.

%I A007578 M1219 #51 Feb 25 2020 02:09:38
%S A007578 1,1,2,4,10,26,76,232,763,2611,9415,35135,136335,544623,2242618,
%T A007578 9463508,40917803,180620411,813405580,3728248990,17377551032,
%U A007578 82232982872,394742985974,1919885633178,9453682648281,47086636037601,237071351741426,1205689994416252
%N A007578 Number of Young tableaux of height <= 7.
%C A007578 Also the number of n-length words w over 7-ary alphabet {a1,a2,...,a7} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a7), where #(z,x) counts the letters x in word z. - _Alois P. Heinz_, May 30 2012
%D A007578 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007578 Alois P. Heinz, <a href="/A007578/b007578.txt">Table of n, a(n) for n = 0..1000</a>
%H A007578 Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
%H A007578 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 A007578 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 A007578 Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, <a href="https://arxiv.org/abs/1708.00513">From Dyck paths to standard Young tableaux</a>, arXiv:1708.00513 [math.CO], 2017.
%H A007578 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A007578 a(n) ~ 45/32 * 7^(n+21/2)/(Pi^(3/2)*n^(21/2)). - _Vaclav Kotesovec_, Sep 11 2013
%p A007578 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
%p A007578       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A007578     end:
%p A007578 g:= proc(n, i, l) option remember;
%p A007578       `if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
%p A007578         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A007578     end:
%p A007578 a:= n-> g(n, 7, []):
%p A007578 seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 10 2012
%p A007578 # second Maple program
%p A007578 a:= proc(n) option remember;
%p A007578       `if`(n<4, [1, 1, 2, 4][n+1],
%p A007578        ((4*n^3+78*n^2+424*n+495)*a(n-1)
%p A007578        +(n-1)*(34*n^2+280*n+305)*a(n-2)
%p A007578        -2*(n-1)*(n-2)*(38*n+145)*a(n-3)
%p A007578        -105*(n-1)*(n-2)*(n-3)*a(n-4)) /
%p A007578        ((n+6)*(n+10)*(n+12)))
%p A007578     end:
%p A007578 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 12 2012
%t A007578 RecurrenceTable[{105 (-3+n) (-2+n) (-1+n) a[-4+n]+2 (-2+n) (-1+n) (145+38 n) a[-3+n]-(-1+n) (305+280 n+34 n^2) a[-2+n]+(-495-424 n-78 n^2-4 n^3) a[-1+n]+(6+n) (10+n) (12+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 A007578 Column k=7 of A182172. - _Alois P. Heinz_, May 30 2012
%K A007578 nonn
%O A007578 0,3
%A A007578 _Simon Plouffe_
%E A007578 More terms from _Alois P. Heinz_, Apr 10 2012