

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 ktuples of positive integers whose elements sum to n.


2



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, 36, 35, 21, 7, 1, 9, 12, 30, 56, 70, 56, 28, 8, 1, 10, 17, 42, 88, 127, 126, 84, 36, 9, 1
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OFFSET

1,2


COMMENTS

C(n,k) counted by A007318(n1,k1) are also called compositions of n of size k (see A181842).


LINKS



EXAMPLE

[1] 1
[2] 2 1
[3] 3 2 1
[4] 4 4 3 1
[5] 5 4 6 4 1
[6] 6 9 11 10 5 1
[7] 7 6 15 20 15 6 1


MAPLE

with(combstruct): # By generating the objects, very inefficient.
a181847_row := proc(n) local k, L, l, R, comp; R := NULL;
for k from 1 to n do
L := 0;
comp := iterstructs(Composition(n), size=k):
while not finished(comp) do
l := nextstruct(comp);
L := L + igcd(op(l));
od;
R := R, L;
od;
R end:
# second Maple program:
with(numtheory):
T := (n, k) > add(phi(d)*binomial(n/d1, k1), d = divisors(n)):


PROG

(Sage) # uses[DivisorTriangle from A327029]
# DivisorTriangle Computes the (0, 0)based version.
DivisorTriangle(euler_phi, lambda n, k: binomial(n1, k1), 10) # Peter Luschny, Aug 27 2019


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



