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%I #25 Feb 15 2023 09:53:22
%S 1,1,2,1,4,6,4,1,10,45,120,210,252,210,120,45,10,1,26,325,2600,14950,
%T 65780,230230,657800,1562275,3124550,5311735,7726160,9657700,10400600,
%U 9657700,7726160,5311735,3124550,1562275,657800,230230,65780,14950,2600,325,26,1
%N Cardinality of the smallest nonempty class of length minimal languages with exactly n nonempty words each over a countably infinite alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
%C a(n) is the smallest nonzero term in column k of A293815.
%H Alois P. Heinz, <a href="/A291057/b291057.txt">Table of n, a(n) for n = 0..1115</a>
%e a(0) = 1: {{}}.
%e a(1) = 1: {{a}}.
%e a(2) = 2: {{a,aa}, {a,ab}}.
%e a(3) = 1: {{a,aa,ab}}.
%e a(4) = 4: {{a,aa,ab,aaa}, {a,aa,ab,aab}, {a,aa,ab,aba}, {a,aa,ab,abc}}.
%e a(5) = 6: {{a,aa,ab,aaa,aab}, {a,aa,ab,aaa,aba}, {a,aa,ab,aaa,abc}, {a,aa,aab,aba}, {a,aa,ab,aab,ab,abc}, {a,aa,ab,aba,abc}}.
%e a(6) = 4: {{a,aa,ab,aaa,aab,aba}, {a,aa,ab,aaa,aab,abc}, {a,aa,ab,aaa,aba,abc}, {a,aa,ab,aab,aba,abc}}.
%e a(7) = 1: {{a,aa,ab,aaa,aab,aba,abc}}.
%e Breaking the sequence into lines after each 1 gives an irregular triangle whose j-th row equals the A000085(j)-th row of A007318 without its leftmost term. The leftmost column of this triangle is A000085:
%e 1;
%e 1;
%e 2, 1;
%e 4, 6, 4, 1;
%e 10, 45, 120, 210, 252, 210, 120, 45, 10, 1;
%e 26, 325, 2600, 14950, 65780, 230230, 657800, 1562275, 3124550, ...
%e ...
%Y Cf. A000085, A007318, A293815, A294129.
%K nonn,look
%O 0,3
%A _Alois P. Heinz_, Oct 20 2017
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