A280275 - OEIS

%I #21 Mar 23 2017 04:15:51

%S 1,1,2,5,12,37,118,387,1312,4445,17034,73339,342532,1616721,7299100,

%T 31195418,129179184,578924785,3057167242,18723356715,120613872016,

%U 738703713245,4080301444740,20353638923275,95273007634552,443132388701107,2149933834972928

%N Number of set partitions of [n] where sizes of distinct blocks are coprime.

%H Alois P. Heinz, <a href="/A280275/b280275.txt">Table of n, a(n) for n = 0..500</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>

%F a(n) = Sum_{k=0..n} A280880(n,k).

%e a(n) = A000110(n) for n<=3.

%e a(4) = 12: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

%e a(5) = 37: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 12|3|4|5, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|23|4|5, 145|2|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.

%p with(numtheory):

%p b:= proc(n, i, s) option remember;

%p       `if`(n=0 or i=1, 1, b(n, i-1, select(x->x<=i-1, s))+

%p       `if`(i>n or factorset(i) intersect s<>{}, 0, b(n-i, i-1,

%p       select(x->x<=i-1, s union factorset(i)))*binomial(n, i)))

%p     end:

%p a:= n-> b(n$2, {}):

%p seq(a(n), n=0..30);

%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 a[n_] := b[n, n, {}] // CoefficientList[#, x]& // Total;

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)

%Y Cf. A000110, A007837, A275313.

%Y Row sums of A280880.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 30 2016