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A277673 Number of n-length words over an n-ary alphabet {a_1,a_2,...,a_n} avoiding consecutive letters a_i, a_{i+1}.

%I A277673 #11 Nov 02 2021 04:35:06
%S A277673 1,1,3,16,136,1547,22012,375231,7445184,168412696,4275561136,
%T A277673 120338946469,3718175865856,125094920949797,4551798150123456,
%U A277673 178094082550301368,7455514741874966528,332495821030327545527,15737024371475868676864,787813565550480151088691
%N A277673 Number of n-length words over an n-ary alphabet {a_1,a_2,...,a_n} avoiding consecutive letters a_i, a_{i+1}.
%H A277673 Alois P. Heinz, <a href="/A277673/b277673.txt">Table of n, a(n) for n = 0..386</a>
%F A277673 a(n) = [x^n] 1/(1+Sum_{j=1..n} (n+1-j)*(-x)^j).
%e A277673 a(3) = 16: 000, 002, 020, 021, 022, 100, 102, 110, 111, 200, 202, 210, 211, 220, 221, 222 (using ternary alphabet {0, 1, 2}).
%p A277673 b:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A277673       -add((-1)^j*(k+1-j)*b(n-j, k), j=1..k)))
%p A277673     end:
%p A277673 a:= n-> b(n$2):
%p A277673 seq(a(n), n=0..25);
%t A277673 b[n_, k_] := b[n, k] = If[n < 0, 0, If[n == 0, 1,
%t A277673      -Sum[(-1)^j (k+1-j) b[n-j, k], {j, 1, k}]]];
%t A277673 a[n_] := b[n, n];
%t A277673 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Nov 02 2021, after _Alois P. Heinz_ *)
%Y A277673 Main diagonal of A277666.
%K A277673 nonn
%O A277673 0,3
%A A277673 _Alois P. Heinz_, Oct 26 2016