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A265583 Array T(n,k) = k*(k-1)^(n-1) read by ascending antidiagonals; k,n>=1. 6
1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 12, 12, 5, 0, 2, 24, 36, 20, 6, 0, 2, 48, 108, 80, 30, 7, 0, 2, 96, 324, 320, 150, 42, 8, 0, 2, 192, 972, 1280, 750, 252, 56, 9, 0, 2, 384, 2916, 5120, 3750, 1512, 392, 72, 10, 0, 2, 768, 8748, 20480, 18750, 9072 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) is the number of n-letter words in a k-letter alphabet with no adjacent letters the same. The factor k represents the number of choices of the first letter, and the n-1 times repeated factor k-1 represents the choices of the next n-1 letters avoiding their predecessor.

The antidiagonal sums are s(d) = 1, 2, 5, 12, 31, 88, 275, 942, 3513, 14158, 61241, 282632, .. for d =n+k >=2.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10011(first 141 antidiagonals, flattened)

FORMULA

T(n,k) = k*A051129(n-1,k-1) = k*A003992(k-1,n-1).

G.f. for column k: k*x/(1-(k-1)*x). - R. J. Mathar, Dec 12 2015

G.f. for array: y/(y-1) - (1+1/x)*y*LerchPhi(y,1,-1/x). - Robert Israel, Dec 13 2018

EXAMPLE

      1       2       3       4       5       6       7

      0       2       6      12      20      30      42

      0       2      12      36      80     150     252

      0       2      24     108     320     750    1512

      0       2      48     324    1280    3750    9072

      0       2      96     972    5120   18750   54432

      0       2     192    2916   20480   93750  326592

T(3,3)=12 counts aba, abc, aca, acb, bab, bac, bca, bcb, cab, cac, cba, cbc. Words like aab or cbb are not counted.

MAPLE

A265583 := proc(n, k)

    k*(k-1)^(n-1) ;

end proc:

seq(seq( A265583(d-k, k), k=1..d-1), d=2..13) ;

MATHEMATICA

T[1, 1] = 1; T[n_, k_] := If[k==1, 0, k*(k-1)^(n-1)]; Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Amiram Eldar, Dec 13 2018 *)

CROSSREFS

Cf. A007283 (column 3), A003946 (column 4), A003947 (column 5), A002378 (row 2), A011379 (row 3), A179824 (row 4), A055897 (diagonal), A265584.

Sequence in context: A209127 A127954 A198061 * A238156 A281260 A182406

Adjacent sequences:  A265580 A265581 A265582 * A265584 A265585 A265586

KEYWORD

nonn,tabl,easy

AUTHOR

R. J. Mathar, Dec 10 2015

STATUS

approved

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Last modified July 24 03:01 EDT 2019. Contains 325290 sequences. (Running on oeis4.)