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%I #18 Oct 18 2018 15:46:06
%S 1,0,1,0,1,1,0,1,3,1,0,1,4,6,1,0,1,15,10,10,1,0,1,6,75,20,15,1,0,1,63,
%T 21,245,35,21,1,0,1,64,476,56,630,56,28,1,0,1,171,540,2100,126,1386,
%U 84,36,1,0,1,130,4185,2640,6930,252,2730,120,45,1
%N Number T(n,k) of set partitions of [n] into exactly k blocks where sizes of distinct blocks are coprime; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A280880/b280880.txt">Rows n = 0..200, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e T(5,1) = 1: 12345.
%e T(5,2) = 15: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
%e T(5,3) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
%e T(5,4) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
%e T(5,5) = 1: 1|2|3|4|5.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 3, 1;
%e 0, 1, 4, 6, 1;
%e 0, 1, 15, 10, 10, 1;
%e 0, 1, 6, 75, 20, 15, 1;
%e 0, 1, 63, 21, 245, 35, 21, 1;
%e 0, 1, 64, 476, 56, 630, 56, 28, 1;
%e 0, 1, 171, 540, 2100, 126, 1386, 84, 36, 1;
%e 0, 1, 130, 4185, 2640, 6930, 252, 2730, 120, 45, 1;
%p with(numtheory):
%p b:= proc(n, i, s) option remember; expand(
%p `if`(n=0 or i=1, x^n, b(n, i-1, select(x->x<=i-1, s))+
%p `if`(i>n or factorset(i) intersect s<>{}, 0, x*b(n-i, i-1,
%p select(x->x<=i-1, s union factorset(i)))*binomial(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..12);
%t b[n_, i_, s_] := b[n, i, s] = Expand[If[n == 0 || i == 1, x^n, b[n, i - 1, Select[s, # <= i - 1 &]] + If[i > n || FactorInteger[i][[All, 1]] ~Intersection~ s != {}, 0, x*b[n - i, i - 1, Select[ s ~Union~ FactorInteger[i][[All, 1]], # <= i - 1 &]]*Binomial[n, i]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 20 2017, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A057427, A194924, A280881, A280882, A280883, A280884, A280885, A280886, A280887, A280888.
%Y T(n+k,n) for k=0-4 give: A000012, A000217, A000292, A051880(n-1) if n>0, A000389(n+4).
%Y Row sums give A280275.
%Y T(2n,n) gives A280889.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Jan 09 2017
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