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A181847 Triangle read by rows: T(n,k)= Sum_{c in C(n,k)}gcd(c) where C(n,k) is the set of all k-tuples of positive integers whose elements sum to n.

%I A181847 #20 Mar 24 2020 12:34:35
%S A181847 1,2,1,3,2,1,4,4,3,1,5,4,6,4,1,6,9,11,10,5,1,7,6,15,20,15,6,1,8,12,24,
%T A181847 36,35,21,7,1,9,12,30,56,70,56,28,8,1,10,17,42,88,127,126,84,36,9,1
%N A181847 Triangle read by rows: T(n,k)= Sum_{c in C(n,k)}gcd(c) where C(n,k) is the set of all k-tuples of positive integers whose elements sum to n.
%C A181847 C(n,k) counted by A007318(n-1,k-1) are also called compositions of n of size k (see A181842).
%e A181847 [1]   1
%e A181847 [2]   2   1
%e A181847 [3]   3   2    1
%e A181847 [4]   4   4    3    1
%e A181847 [5]   5   4    6    4    1
%e A181847 [6]   6   9   11   10    5   1
%e A181847 [7]   7   6   15   20   15   6   1
%p A181847 with(combstruct): # By generating the objects, very inefficient.
%p A181847 a181847_row := proc(n) local k,L,l,R,comp; R := NULL;
%p A181847 for k from 1 to n do
%p A181847    L := 0;
%p A181847    comp := iterstructs(Composition(n),size=k):
%p A181847    while not finished(comp) do
%p A181847       l := nextstruct(comp);
%p A181847       L := L + igcd(op(l));
%p A181847    od;
%p A181847    R := R,L;
%p A181847 od;
%p A181847 R end:
%p A181847 # second Maple program:
%p A181847 with(numtheory):
%p A181847 T := (n, k) -> add(phi(d)*binomial(n/d-1, k-1), d = divisors(n)):
%p A181847 seq(seq(T(n, k), k=1..n), n=1..10); # _Peter Luschny_, Aug 27 2019
%o A181847 (Sage) # uses[DivisorTriangle from A327029]
%o A181847 # DivisorTriangle Computes the (0,0)-based version.
%o A181847 DivisorTriangle(euler_phi, lambda n,k: binomial(n-1, k-1), 10) # _Peter Luschny_, Aug 27 2019
%Y A181847 Cf. A034738, A065567, A065568, A327029.
%K A181847 nonn,tabl
%O A181847 1,2
%A A181847 _Peter Luschny_, Dec 07 2010