OFFSET
0,3
COMMENTS
Also the number of n-length words w over 6-ary alphabet {a1,a2,...,a6} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a6), where #(z,x) counts the letters x in word z. - Alois P. Heinz, May 30 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Preprint. (Annotated scanned copy)
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.
Alon Regev, Amitai Regev, Doron Zeilberger, Identities in character tables of S_n, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
FORMULA
a(n) ~ 3/4 * 6^(n+15/2)/(Pi^(3/2)*n^(15/2)). - Vaclav Kotesovec, Sep 11 2013
D-finite with recurrence +(n+5)*(n+9)*(n+8)*a(n) +4*(-5*n^2-46*n-84)*a(n-1) -4*(n-1)*(10*n^2+58*n+33)*a(n-2) +144*(n-1)*(n-2)*a(n-3) +144*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 23 2021
MAPLE
h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) option remember;
`if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
end:
a:= n-> g(n, 6, []):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 18 2012
# second Maple program:
a:= proc(n) option remember;
`if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*a(n-1)
+4*(n-1)*(10*n^2+58*n+33)*a(n-2) -144*(n-1)*(n-2)*a(n-3)
-144*(n-1)*(n-2)*(n-3)*a(n-4))/ ((n+5)*(n+8)*(n+9)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 12 2012
MATHEMATICA
RecurrenceTable[{144 (-3+n) (-2+n) (-1+n) a[-4+n]+144 (-2+n) (-1+n) a[-3+n]-4 (-1+n) (33+58 n+10 n^2) a[-2+n]-4 (84+46 n+5 n^2) a[-1+n]+(5+n) (8+n) (9+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10}, a, {n, 20}] (* Vaclav Kotesovec, Sep 11 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Apr 10 2012
STATUS
approved