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.

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, 7, 0, 42, 492, 864, 580, 210, 49, 8, 0, 68, 1344, 3276, 2800, 1230, 336, 64, 9, 0, 110, 3672, 12420, 13520, 7200, 2310, 504, 81, 10, 0, 178, 10032, 47088, 65280, 42150, 15876, 3976, 720, 100, 11

Offset: 1

Author

R. J. Mathar, Dec 10 2015

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			1      2      3      4      5       6       7        8
1      4      9     16     25      36      49       64
0      6     24     60    120     210     336      504
0     10     66    228    580    1230    2310     3976
0     16    180    864   2800    7200   15876    31360
0     26    492   3276  13520   42150  109116   247352
0     42   1344  12420  65280  246750  749952  1950984
0     68   3672  47088 315200 1444500 5154408 15388352
T(3,2) =6 counts the 3-letter words aab, aba, abb, bba, bab, baa. The words aaa and bbb are not counted.
		

Crossrefs

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).

Programs

Maple

A265584 := proc(n,k)
    (1+x+x^2)/(1-(k-1)*x-(k-1)*x^2) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(seq( A265584(d-k,k),k=1..d-1),d=2..13) ;

Mathematica

T[n_, k_] := SeriesCoefficient[(1+x+x^2)/(1-(k-1)*x-(k-1)*x^2), {x, 0, n}];
Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Mar 26 2020, from Maple *)

Formula

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

T(5,k) = k^2*(k+2)*(k-1)^2.

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

T(7,k) = k*(k+1)*(k^2+2*k-1)*(k-1)^3.