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A257740
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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.
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16
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144074(n,k-i).
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000041 (for n>0), A261043, A320213, A320214, A320215, A320216, A320217, A320218, A320219, A320220.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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