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A280275 Number of set partitions of [n] where sizes of distinct blocks are coprime. 3
1, 1, 2, 5, 12, 37, 118, 387, 1312, 4445, 17034, 73339, 342532, 1616721, 7299100, 31195418, 129179184, 578924785, 3057167242, 18723356715, 120613872016, 738703713245, 4080301444740, 20353638923275, 95273007634552, 443132388701107, 2149933834972928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

Wikipedia, Coprime integers

Wikipedia, Partition of a set

FORMULA

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

EXAMPLE

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

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.

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.

MAPLE

with(numtheory):

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

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

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

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

    end:

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

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

MATHEMATICA

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]]]];

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

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Mar 23 2017, translated from Maple *)

CROSSREFS

Cf. A000110, A007837, A275313.

Row sums of A280880.

Sequence in context: A267399 A267400 A052302 * A009598 A002216 A024717

Adjacent sequences:  A280272 A280273 A280274 * A280276 A280277 A280278

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 30 2016

STATUS

approved

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Last modified December 11 21:00 EST 2019. Contains 329937 sequences. (Running on oeis4.)