Ax A265624 := proc(n, k)

seq(seq(AxA265624(d-k, k), k=1..d-1), d=2..10) ;

allocated for R. J. Mathar

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

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, 7, 0, 88, 666, 996, 620, 216, 49, 8, 0, 162, 1944, 3936, 3080, 1290, 343, 64, 9, 0, 298, 5676, 15552, 15300, 7710, 2394, 512, 81, 10, 0, 548, 16572, 61452, 76000, 46080, 16716, 4088, 729, 100, 11, 0, 1008, 48384, 242820

1,3

T(2,k)=k^2.

T(3,k)=k^3.

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

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

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

G.f of row k: k*x*(1+x+x^2)/(1-(k-1)*x-(k-1)*x^2-(k-1)*x^3).

1 2 3 4 5 6 7 8

1 4 9 16 25 36 49 64

1 8 27 64 125 216 343 512

0 14 78 252 620 1290 2394 4088

0 26 228 996 3080 7710 16716 32648

0 48 666 3936 15300 46080 116718 260736

0 88 1944 15552 76000 275400 814968 2082304

0 162 5676 61452 377520 1645950 5690412 16629816

Ax := proc(n, k)

local x;

k*x*(1+x+x^2)/(1-(k-1)*x-(k-1)*x^2-(k-1)*x^3) ;

coeftayl(%, x=0, n) ;

end proc;

seq(seq(Ax(d-k, k), k=1..d-1), d=2..10) ;

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

allocated

nonn,tabl,easy

R. J. Mathar, Dec 10 2015

approved

editing

allocated for R. J. Mathar

allocated

approved