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

1,3

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.

Links

Table of n, a(n) for n=1..66.

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.

Example

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.

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 *)

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).

Sequence in context: A192423 A368667 A359214 * A360205 A078909 A067458

Adjacent sequences: A265581 A265582 A265583 * A265585 A265586 A265587

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Dec 10 2015

Status

approved