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A280880 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.

%I A280880 #18 Oct 18 2018 15:46:06
%S A280880 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 A280880 21,245,35,21,1,0,1,64,476,56,630,56,28,1,0,1,171,540,2100,126,1386,
%U A280880 84,36,1,0,1,130,4185,2640,6930,252,2730,120,45,1
%N A280880 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 A280880 Alois P. Heinz, <a href="/A280880/b280880.txt">Rows n = 0..200, flattened</a>
%H A280880 Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>
%H A280880 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A280880 T(5,1) = 1: 12345.
%e A280880 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 A280880 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 A280880 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 A280880 T(5,5) = 1: 1|2|3|4|5.
%e A280880 Triangle T(n,k) begins:
%e A280880   1;
%e A280880   0, 1;
%e A280880   0, 1,   1;
%e A280880   0, 1,   3,    1;
%e A280880   0, 1,   4,    6,    1;
%e A280880   0, 1,  15,   10,   10,    1;
%e A280880   0, 1,   6,   75,   20,   15,    1;
%e A280880   0, 1,  63,   21,  245,   35,   21,    1;
%e A280880   0, 1,  64,  476,   56,  630,   56,   28,   1;
%e A280880   0, 1, 171,  540, 2100,  126, 1386,   84,  36,  1;
%e A280880   0, 1, 130, 4185, 2640, 6930,  252, 2730, 120, 45, 1;
%p A280880 with(numtheory):
%p A280880 b:= proc(n, i, s) option remember; expand(
%p A280880       `if`(n=0 or i=1, x^n, b(n, i-1, select(x->x<=i-1, s))+
%p A280880       `if`(i>n or factorset(i) intersect s<>{}, 0, x*b(n-i, i-1,
%p A280880       select(x->x<=i-1, s union factorset(i)))*binomial(n, i))))
%p A280880     end:
%p A280880 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, {})):
%p A280880 seq(T(n), n=0..12);
%t A280880 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 A280880 Columns k=0-10 give: A000007, A057427, A194924, A280881, A280882, A280883, A280884, A280885, A280886, A280887, A280888.
%Y A280880 T(n+k,n) for k=0-4 give: A000012, A000217, A000292, A051880(n-1) if n>0, A000389(n+4).
%Y A280880 Row sums give A280275.
%Y A280880 T(2n,n) gives A280889.
%K A280880 nonn,tabl
%O A280880 0,9
%A A280880 _Alois P. Heinz_, Jan 09 2017