A265583 Array T(n,k) = k*(k-1)^(n-1) read by ascending antidiagonals; k,n >= 1.

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, 2744, 576, 90, 11

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 *)

Prog

(PARI) T(n, k) = if(n==k==1, 1, k*(k-1)^(n-k-1) );
for(n=2, 15, for(k=1, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 10 2019
(Magma)
T:= func< n, k | (n eq 1 and k eq 1) select 1 else k*(k-1)^(n-k-1) >;
[T(n, k): k in [1..n-1], n in [2..15]]; // G. C. Greubel, Aug 10 2019
(Sage)
def T(n, k):
    if (n==k==1): return 1
    else: return k*(k-1)^(n-k-1)
[[T(n, k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Aug 10 2019
(GAP)
T:= function(n, k)
    if (n=1 and k=1) then return 1;
    else return k*(k-1)^(n-k-1);
    fi;
  end;
Flat(List([2..15], n-> List([1..n-1], k-> T(n, k) ))); # G. C. Greubel, Aug 10 2019

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 * A339754 A238156 A281260

Adjacent sequences: A265580 A265581 A265582 * A265584 A265585 A265586

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Dec 10 2015

Status

approved