%I #27 Feb 14 2021 13:59:41

%S 1,0,1,0,2,3,0,3,14,13,0,5,49,114,73,0,7,148,672,1028,501,0,11,427,

%T 3334,9182,10310,4051,0,15,1170,15030,66584,129485,114402,37633,0,22,

%U 3150,63978,428653,1285815,1918083,1394414,394353,0,30,8288,261880,2557972,11117600,24917060,30044014,18536744,4596553

%N 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.

%C 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.

%H Alois P. Heinz, <a href="/A257740/b257740.txt">Rows n = 0..140, flattened</a>

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

%e T(2,2) = 3: {ab}, {ba}, {a,b}.

%e 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}.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 2, 3;

%e 0, 3, 14, 13;

%e 0, 5, 49, 114, 73;

%e 0, 7, 148, 672, 1028, 501;

%e 0, 11, 427, 3334, 9182, 10310, 4051;

%e 0, 15, 1170, 15030, 66584, 129485, 114402, 37633;

%e 0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353;

%e ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(

%p d*k^d, d=numtheory[divisors](j)) *A(n-j, k), j=1..n)/n)

%p end:

%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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 *)

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

%Y Row sums give A257741.

%Y Main diagonal gives A000262.

%Y T(2n,n) gives A257742.

%Y Cf. A144074, A319501.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, May 06 2015

%E Name changed by _Alois P. Heinz_, Sep 21 2018