A265584 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

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