cp's OEIS Frontend

A065567 T(n,m) is the sum over all m-subsets of {1,...,n} of the gcd of the subset.

%I A065567 #33 Sep 05 2023 20:46:27
%S A065567 1,3,1,6,3,1,10,7,4,1,15,11,10,5,1,21,20,21,15,6,1,28,26,36,35,21,7,1,
%T A065567 36,38,60,71,56,28,8,1,45,50,90,127,126,84,36,9,1,55,67,132,215,253,
%U A065567 210,120,45,10,1,66,77,177,335,463,462,330,165,55,11,1,78,105,250,512,798,925,792,495,220,66,12,1
%N A065567 T(n,m) is the sum over all m-subsets of {1,...,n} of the gcd of the subset.
%C A065567 First differences of row sums equals A034738.
%H A065567 Alois P. Heinz, <a href="/A065567/b065567.txt">Rows n = 1..200</a> (first 31 rows from Sean A. Irvine)
%F A065567 Sum_{k=1..n} (-1)^(k+1) * T(n,k) = A002088(n). - _Alois P. Heinz_, Sep 05 2023
%e A065567 Triangle begins:
%e A065567    1;
%e A065567    3,  1;
%e A065567    6,  3,  1;
%e A065567   10,  7,  4, 1;
%e A065567   15, 11, 10, 5, 1;
%e A065567   ...
%e A065567 T(4,2) = 7 = gcd(1,2) + gcd(1,3) + gcd(1,4) + gcd(2,3) + gcd(2,4) + gcd(3,4).
%p A065567 with(combstruct):
%p A065567 a065567_row := proc(n) local k,L,l,R,comb;
%p A065567 R := NULL;
%p A065567 for k from 1 to n do
%p A065567    L := 0;
%p A065567    comb := iterstructs(Combination(n),size=k):
%p A065567    while not finished(comb) do
%p A065567       l := nextstruct(comb);
%p A065567       L := L + igcd(op(l));
%p A065567    od;
%p A065567    R := R,L;
%p A065567 od;
%p A065567 R end: # _Peter Luschny_, Dec 07 2010
%p A065567 # second Maple program:
%p A065567 b:= proc(n, g, t) option remember; `if`(n=0, g*x^t,
%p A065567       b(n-1, igcd(g, n), t+1)+b(n-1, g, t))
%p A065567     end:
%p A065567 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0$2)):
%p A065567 seq(T(n), n=1..12);  # _Alois P. Heinz_, Sep 05 2023
%t A065567 Table[Plus@@(GCD@@@KSubsets[Range[n], m]), {n, 16}, {m, n}]
%Y A065567 Row sums give A065568.
%Y A065567 T(2n,n) gives A244174 for n>=1.
%Y A065567 T(2n,n+1) gives A001791 for n>=1.
%Y A065567 T(2n+1,n+1) gives A001700 for n>=0.
%Y A065567 Cf. A002088, A034738.
%K A065567 nonn,tabl
%O A065567 1,2
%A A065567 _Wouter Meeussen_, Nov 30 2001