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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.

%I A257740 #27 Feb 14 2021 13:59:41
%S A257740 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 A257740 3334,9182,10310,4051,0,15,1170,15030,66584,129485,114402,37633,0,22,
%U A257740 3150,63978,428653,1285815,1918083,1394414,394353,0,30,8288,261880,2557972,11117600,24917060,30044014,18536744,4596553
%N 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.
%C A257740 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 A257740 Alois P. Heinz, <a href="/A257740/b257740.txt">Rows n = 0..140, flattened</a>
%F A257740 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144074(n,k-i).
%e A257740 T(2,2) = 3: {ab}, {ba}, {a,b}.
%e A257740 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 A257740 Triangle T(n,k) begins:
%e A257740   1;
%e A257740   0,  1;
%e A257740   0,  2,    3;
%e A257740   0,  3,   14,    13;
%e A257740   0,  5,   49,   114,     73;
%e A257740   0,  7,  148,   672,   1028,     501;
%e A257740   0, 11,  427,  3334,   9182,   10310,    4051;
%e A257740   0, 15, 1170, 15030,  66584,  129485,  114402,   37633;
%e A257740   0, 22, 3150, 63978, 428653, 1285815, 1918083, 1394414, 394353;
%e A257740   ...
%p A257740 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A257740       d*k^d, d=numtheory[divisors](j)) *A(n-j, k), j=1..n)/n)
%p A257740     end:
%p A257740 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A257740 seq(seq(T(n, k), k=0..n), n=0..10);
%t A257740 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 A257740 Columns k=0-10 give: A000007, A000041 (for n>0), A261043, A320213, A320214, A320215, A320216, A320217, A320218, A320219, A320220.
%Y A257740 Row sums give A257741.
%Y A257740 Main diagonal gives A000262.
%Y A257740 T(2n,n) gives A257742.
%Y A257740 Cf. A144074, A319501.
%K A257740 nonn,tabl
%O A257740 0,5
%A A257740 _Alois P. Heinz_, May 06 2015
%E A257740 Name changed by _Alois P. Heinz_, Sep 21 2018