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%I #11 Sep 02 2014 13:46:54
%S 1,24,105,2575,115955,7364321,586368681,54862627919,5795673908453,
%T 673174876488400,84386541996407430,11262879538848476760,
%U 1584243362361105791448,233004893382083549345048,35610340402841609968092950,5627093485549459958456588775
%N Number of words w where each letter of the 4-ary alphabet occurs n times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.
%H Alois P. Heinz, <a href="/A213874/b213874.txt">Table of n, a(n) for n = 0..140</a>
%F For n > 1, a(n) = 8*(9297776*n^10 + 17051200*n^9 - 11545329*n^8 - 20688255*n^7 + 7760028*n^6 + 7548270*n^5 - 2879537*n^4 - 619195*n^3 + 326046*n^2 - 30420*n + 216) * (4*n-5)! / (3 * (2*n-1) * (2*n+1) * (2*n+3) * (9*n^2-9*n+2) * (9*n^2+9*n+2) * (n-2)! * (n+1)! * (n+2)! * (n+3)!). - _Vaclav Kotesovec_, Sep 02 2014
%e a(0) = 1: the empty word.
%e a(1) = 24: abcd, abdc, ..., dcab, dcba, (all permutations of 4 letters).
%e a(2) = 105: aabbccdd, aabbcdcd, aabbdccd, ..., dcaabbcd, dcababcd, dcbaabcd.
%t Flatten[{1,24,Table[8*(9297776*n^10 + 17051200*n^9 - 11545329*n^8 - 20688255*n^7 + 7760028*n^6 + 7548270*n^5 - 2879537*n^4 - 619195*n^3 + 326046*n^2 - 30420*n + 216) * (4*n-5)! / (3 * (2*n-1) * (2*n+1) * (2*n+3) * (9*n^2-9*n+2) * (9*n^2+9*n+2) * (n-2)! * (n+1)! * (n+2)! * (n+3)!),{n,2,20}]}] (* _Vaclav Kotesovec_, Sep 02 2014 *)
%Y Column k=4 of A213275.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Jun 23 2012
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