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A265624 Array T(n,k): The number of words of length n in an alphabet of size k which do not contain 4 consecutive letters.

%I A265624 #13 Dec 12 2015 08:06:46
%S A265624 1,1,2,1,4,3,0,8,9,4,0,14,27,16,5,0,26,78,64,25,6,0,48,228,252,125,36,
%T A265624 7,0,88,666,996,620,216,49,8,0,162,1944,3936,3080,1290,343,64,9,0,298,
%U A265624 5676,15552,15300,7710,2394,512,81,10,0,548,16572,61452
%N A265624 Array T(n,k): The number of words of length n in an alphabet of size k which do not contain 4 consecutive letters.
%F A265624 T(2,k) = k^2.
%F A265624 T(3,k) = k^3.
%F A265624 T(4,k) = k*(k+1)*(k^2+3*k+3).
%F A265624 T(5,k) = k*(k+1)*(k^3+4*k^2+6*k+2).
%F A265624 T(6,k) = k*(k+1)^2*(k^3+4*k^2+6*k+1).
%F A265624 G.f. of row k: k*x*(1+x+x^2)/(1+(1-k)*x*(x^2+x+1)).
%e A265624   1    2      3      4      5       6       7        8
%e A265624   1    4      9     16     25      36      49       64
%e A265624   1    8     27     64    125     216     343      512
%e A265624   0   14     78    252    620    1290    2394     4088
%e A265624   0   26    228    996   3080    7710   16716    32648
%e A265624   0   48    666   3936  15300   46080  116718   260736
%e A265624   0   88   1944  15552  76000  275400  814968  2082304
%e A265624   0  162   5676  61452 377520 1645950 5690412 16629816
%p A265624 A265624 := proc(n,k)
%p A265624         local x;
%p A265624         k*x*(1+x+x^2)/(1+(1-k)*x*(x^2+x+1)) ;
%p A265624         coeftayl(%,x=0,n) ;
%p A265624 end proc;
%p A265624 seq(seq(A265624(d-k,k),k=1..d-1),d=2..10) ;
%Y A265624 Cf. A135491 (column k=2), A181137 (k=3), A188714 (k=4), A265583 (not 2 consecutive letters), A265584 (not 3 consecutive letters).
%K A265624 nonn,tabl,easy
%O A265624 1,3
%A A265624 _R. J. Mathar_, Dec 10 2015