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 A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 1, 3, 6, 5, 0, 1, 1, 3, 7, 15, 7, 0, 1, 1, 3, 7, 19, 31, 11, 0, 1, 1, 3, 7, 20, 48, 73, 15, 0, 1, 1, 3, 7, 20, 53, 131, 155, 22, 0, 1, 1, 3, 7, 20, 54, 157, 348, 351, 30, 0, 1, 1, 3, 7, 20, 54, 163, 455, 954, 755, 42, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Antidiagonals n = 0..80, flattened FORMULA G.f. of column k: Product_{j>=1} 1/(1-x^j)^A182172(j,k). A(n,k) = Sum_{j=0..k} A293109(n,j). A(n,n) = A(n,k) for all k >= n. EXAMPLE Square array A(n,k) begins:   1,  1,   1,   1,   1,   1,   1,   1, ...   0,  1,   1,   1,   1,   1,   1,   1, ...   0,  2,   3,   3,   3,   3,   3,   3, ...   0,  3,   6,   7,   7,   7,   7,   7, ...   0,  5,  15,  19,  20,  20,  20,  20, ...   0,  7,  31,  48,  53,  54,  54,  54, ...   0, 11,  73, 131, 157, 163, 164, 164, ...   0, 15, 155, 348, 455, 492, 499, 500, ... MAPLE h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]     n, 0, g(n-i, i, [l[], i])))))     end: A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])       *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)     end: seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA h[l_] := Function [n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] < j, 0, 1], {k, i+1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]]; g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j] }]*A[n - j, k], {j, 1, n}]/n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *) CROSSREFS Columns k=0-10 give: A000007, A000041, A293732, A293733, A293734, A293735, A293736, A293737, A293738, A293739, A293740. Main diagonal gives A293110. Cf. A182172, A293109, A293112. Sequence in context: A219782 A035788 A097559 * A172237 A246181 A123226 Adjacent sequences:  A293105 A293106 A293107 * A293109 A293110 A293111 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 30 2017 STATUS approved

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Last modified May 21 19:10 EDT 2019. Contains 323444 sequences. (Running on oeis4.)