|
%I M1219
%S 1,2,4,10,26,76,232,763,2611,9415,35135,136335,544623,2242618,9463508,
%T 40917803,180620411,813405580,3728248990,17377551032,82232982872,
%U 394742985974,1919885633178,9453682648281,47086636037601,237071351741426,1205689994416252
%N Number of Young tableaux of height <= 7.
%C 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 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="/A007578/b007578.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, 7, []):
%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 ((4*n^3+78*n^2+424*n+495)*a(n-1)
%p +(n-1)*(34*n^2+280*n+305)*a(n-2)
%p -2*(n-1)*(n-2)*(38*n+145)*a(n-3)
%p -105*(n-1)*(n-2)*(n-3)*a(n-4)) /
%p ((n+6)*(n+10)*(n+12)))
%p end:
%p seq (a(n), n=1..30); # _Alois P. Heinz_, Oct 12 2012
%Y Column k=7 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
|
