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%I #19 Feb 15 2023 09:56:07
%S 1,0,1,0,2,0,4,2,0,10,5,0,26,18,1,0,76,52,8,0,232,168,30,0,764,533,
%T 114,4,0,2620,1792,411,22,0,9496,6161,1462,116,0,35696,22088,5237,482,
%U 6,0,140152,81690,18998,1966,48,0,568504,313224,70220,7682,274
%N Number T(n,k) of sets of exactly k nonempty words with a total of n letters over n-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; triangle T(n,k), n>=0, read by rows.
%C The smallest nonzero term in column k is A291057(k).
%H Alois P. Heinz, <a href="/A293815/b293815.txt">Rows n = 0..300, flattened</a>
%F G.f.: Product_{j>=1} (1+y*x^j)^A000085(j).
%e T(0,0) = 1: {}.
%e T(3,1) = 4: {aaa}, {aab}, {aba}, {abc}.
%e T(3,2) = 2: {a,aa}, {a,ab}.
%e T(4,2) = 5: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,ab}.
%e T(5,3) = 1: {a,aa,ab}.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2;
%e 0, 4, 2;
%e 0, 10, 5;
%e 0, 26, 18, 1;
%e 0, 76, 52, 8;
%e 0, 232, 168, 30;
%e 0, 764, 533, 114, 4;
%e 0, 2620, 1792, 411, 22;
%e 0, 9496, 6161, 1462, 116;
%e 0, 35696, 22088, 5237, 482, 6;
%e 0, 140152, 81690, 18998, 1966, 48;
%e ...
%p g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..15);
%t g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1]* Binomial[g[i], j]*x^j, {j, 0, n/i}]]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
%t Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Jun 04 2018, from Maple *)
%Y Columns k=0-10 give: A000007, A000085 (for n>0), A293964, A293965, A293966, A293967, A293968, A293969, A293970, A293971, A293972.
%Y Row sums give A293114.
%Y Cf. A208741, A293808, A291057, A294129.
%K nonn,tabf
%O 0,5
%A _Alois P. Heinz_, Oct 16 2017
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