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

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, 36, 38, 60, 71, 56, 28, 8, 1, 45, 50, 90, 127, 126, 84, 36, 9, 1, 55, 67, 132, 215, 253, 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

Offset

1,2

Comments

First differences of row sums equals A034738.

Links

Alois P. Heinz, Rows n = 1..200 (first 31 rows from Sean A. Irvine)

Formula

Sum_{k=1..n} (-1)^(k+1) * T(n,k) = A002088(n). - Alois P. Heinz, Sep 05 2023

Example

Triangle begins:
   1;
   3,  1;
   6,  3,  1;
  10,  7,  4, 1;
  15, 11, 10, 5, 1;
  ...
T(4,2) = 7 = gcd(1,2) + gcd(1,3) + gcd(1,4) + gcd(2,3) + gcd(2,4) + gcd(3,4).

Maple

with(combstruct):
a065567_row := proc(n) local k, L, l, R, comb;
R := NULL;
for k from 1 to n do
   L := 0;
   comb := iterstructs(Combination(n), size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := L + igcd(op(l));
   od;
   R := R, L;
od;
R end: # Peter Luschny, Dec 07 2010
# second Maple program:
b:= proc(n, g, t) option remember; `if`(n=0, g*x^t,
      b(n-1, igcd(g, n), t+1)+b(n-1, g, t))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Sep 05 2023

Mathematica

Table[Plus@@(GCD@@@KSubsets[Range[n], m]), {n, 16}, {m, n}]

Crossrefs

Row sums give A065568.

T(2n,n) gives A244174 for n>=1.

T(2n,n+1) gives A001791 for n>=1.

T(2n+1,n+1) gives A001700 for n>=0.

Cf. A002088, A034738.

Sequence in context: A122432 A131110 A133093 * A100861 A131031 A130452

Adjacent sequences: A065564 A065565 A065566 * A065568 A065569 A065570

Keyword

nonn,tabl

Author

Wouter Meeussen, Nov 30 2001

Status

approved