A257740 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 3, 0, 3, 14, 13, 0, 5, 49, 114, 73, 0, 7, 148, 672, 1028, 501, 0, 11, 427, 3334, 9182, 10310, 4051, 0, 15, 1170, 15030, 66584, 129485, 114402, 37633, 0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353, 0, 30, 8288, 261880, 2557972, 11117600, 24917060, 30044014, 18536744, 4596553

Offset: 0

Author

Alois P. Heinz, May 06 2015

Keywords

• nonn
• tabl

Comments

Row n is the inverse binomial transform of the n-th row of array A144074, which has the Euler transform of the powers of k in column k.

Examples

			T(2,2) = 3: {ab}, {ba}, {a,b}.
T(3,2) = 14: {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {a,ab}, {a,ba}, {a,bb}, {aa,b}, {ab,b}, {b,ba}, {a,a,b}, {a,b,b}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    3;
  0,  3,   14,    13;
  0,  5,   49,   114,     73;
  0,  7,  148,   672,   1028,     501;
  0, 11,  427,  3334,   9182,   10310,    4051;
  0, 15, 1170, 15030,  66584,  129485,  114402,   37633;
  0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A000041 (for n>0), A261043, A320213, A320214, A320215, A320216, A320217, A320218, A320219, A320220.

Row sums give A257741.

Main diagonal gives A000262.

T(2n,n) gives A257742.

Cf. A144074, A319501.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*k^d, d=numtheory[divisors](j)) *A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*k^#&]*A[n - j, k], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 23 2017, adapted from Maple *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144074(n,k-i).

Extensions

Name changed by Alois P. Heinz, Sep 21 2018