%I #13 Dec 12 2015 08:06:46

%S 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 7,0,88,666,996,620,216,49,8,0,162,1944,3936,3080,1290,343,64,9,0,298,

%U 5676,15552,15300,7710,2394,512,81,10,0,548,16572,61452

%N 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 T(2,k) = k^2.

%F T(3,k) = k^3.

%F T(4,k) = k*(k+1)*(k^2+3*k+3).

%F T(5,k) = k*(k+1)*(k^3+4*k^2+6*k+2).

%F T(6,k) = k*(k+1)^2*(k^3+4*k^2+6*k+1).

%F G.f. of row k: k*x*(1+x+x^2)/(1+(1-k)*x*(x^2+x+1)).

%e 1 2 3 4 5 6 7 8

%e 1 4 9 16 25 36 49 64

%e 1 8 27 64 125 216 343 512

%e 0 14 78 252 620 1290 2394 4088

%e 0 26 228 996 3080 7710 16716 32648

%e 0 48 666 3936 15300 46080 116718 260736

%e 0 88 1944 15552 76000 275400 814968 2082304

%e 0 162 5676 61452 377520 1645950 5690412 16629816

%p A265624 := proc(n,k)

%p local x;

%p k*x*(1+x+x^2)/(1+(1-k)*x*(x^2+x+1)) ;

%p coeftayl(%,x=0,n) ;

%p end proc;

%p seq(seq(A265624(d-k,k),k=1..d-1),d=2..10) ;

%Y Cf. A135491 (column k=2), A181137 (k=3), A188714 (k=4), A265583 (not 2 consecutive letters), A265584 (not 3 consecutive letters).

%K nonn,tabl,easy

%O 1,3

%A _R. J. Mathar_, Dec 10 2015