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A265584 Array T(n,k) counting words with n letters drawn from a k-letter alphabet with no letter appearing thrice in a 3-letter subword.

%I A265584 #17 Mar 26 2020 07:06:14
%S A265584 1,1,2,0,4,3,0,6,9,4,0,10,24,16,5,0,16,66,60,25,6,0,26,180,228,120,36,
%T A265584 7,0,42,492,864,580,210,49,8,0,68,1344,3276,2800,1230,336,64,9,0,110,
%U A265584 3672,12420,13520,7200,2310,504,81,10,0,178,10032,47088,65280,42150,15876,3976,720,100,11
%N A265584 Array T(n,k) counting words with n letters drawn from a k-letter alphabet with no letter appearing thrice in a 3-letter subword.
%C A265584 The antidiagonal sums are s(d) = 1, 3, 7, 19, 55, 173, 597, 2245, 9127, 39827, 185411, 916177, 4784217,.. at index d=n+k >=2.
%F A265584 T(4,k) = k*(k-1)*(k^2+k-1).
%F A265584 T(5,k) = k^2*(k+2)*(k-1)^2.
%F A265584 T(6,k) = k*(k^3+2*k^2-k-1)*(k-1)^2.
%F A265584 T(7,k) = k*(k+1)*(k^2+2*k-1)*(k-1)^3.
%e A265584 1      2      3      4      5       6       7        8
%e A265584 1      4      9     16     25      36      49       64
%e A265584 0      6     24     60    120     210     336      504
%e A265584 0     10     66    228    580    1230    2310     3976
%e A265584 0     16    180    864   2800    7200   15876    31360
%e A265584 0     26    492   3276  13520   42150  109116   247352
%e A265584 0     42   1344  12420  65280  246750  749952  1950984
%e A265584 0     68   3672  47088 315200 1444500 5154408 15388352
%e A265584 T(3,2) =6 counts the 3-letter words aab, aba, abb, bba, bab, baa. The words aaa and bbb are not counted.
%p A265584 A265584 := proc(n,k)
%p A265584     (1+x+x^2)/(1-(k-1)*x-(k-1)*x^2) ;
%p A265584     coeftayl(%,x=0,n) ;
%p A265584 end proc:
%p A265584 seq(seq( A265584(d-k,k),k=1..d-1),d=2..13) ;
%t A265584 T[n_, k_] := SeriesCoefficient[(1+x+x^2)/(1-(k-1)*x-(k-1)*x^2), {x, 0, n}];
%t A265584 Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Mar 26 2020, from Maple *)
%Y A265584 Cf. A265583 (no letter twice), A265624. A000290 (row 2), A007531 (row 3), A006355 (column 2), A121907 (column 3), A123620 (column 4), A123871 (column 5), A123887 (column 6).
%K A265584 nonn,tabl,easy
%O A265584 1,3
%A A265584 _R. J. Mathar_, Dec 10 2015