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

0,5

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.

Links

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

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A261719 A137663 A370983 * A161628 A244119 A122059

Adjacent sequences: A257737 A257738 A257739 * A257741 A257742 A257743

Keyword

nonn,tabl

Author

Alois P. Heinz, May 06 2015

Extensions

Name changed by Alois P. Heinz, Sep 21 2018

Status

approved