A181853 Triangle read by rows: T(n,k) = Sum_{c in C(n,k)} lcm(c) where C(n,k) is the set of all k-subsets of {1,2,...,n}.

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 31, 34, 12, 1, 15, 81, 189, 182, 60, 1, 21, 141, 393, 494, 282, 60, 1, 28, 288, 1380, 3245, 3740, 2034, 420, 1, 36, 456, 2716, 8293, 13268, 11338, 4908, 840, 1, 45, 726, 5578, 22207, 47351, 57598, 40602, 15564, 2520

Offset

0,5

Comments

The C(n,k) are also called combinations of n with size k (see A181842).

Main diagonal gives: A003418. Lower diagonal gives: A094308. Column k=1 gives: A000217. - Alois P. Heinz, Jul 29 2013

Links

Alois P. Heinz, Rows n = 0..46, flattened

Example

[0]   1
[1]   1    1
[2]   1    3     2
[3]   1    6    11     6
[4]   1   10    31    34    12
[5]   1   15    81   189   182    60
[6]   1   21   141   393   494   282   60

Maple

with(combstruct):
a181853_row := proc(n) local k, L, l, R, comb;
R := NULL;
for k from 0 to n do
   L := 0;
   comb := iterstructs(Combination(n), size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := L + ilcm(op(l));
   od;
   R := R, L;
od;
R end:
# second Maple program:
b:= proc(n, k) option remember; `if`(k=0, [1],
     [`if`(k<n, b(n-1, k), [])[], seq(ilcm(c, n), c=b(n-1, k-1))])
    end:
T:= (n, k)-> add(c, c=b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 29 2013
# third Maple program:
b:= proc(n, m) option remember; expand(`if`(n=0, m,
      b(n-1, ilcm(m, n))*x+b(n-1, m)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Sep 05 2023

Mathematica

t[_, 0] = 1; t[n_, k_] := Sum[LCM @@ c, {c, Subsets[Range[n], {k}]}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Prog

(Sage) # (After Alois P. Heinz)
@CachedFunction
def b(n, k):
    if k == 0: return [1]
    w = b(n-1, k) if k<n else [0]
    return w + [lcm(c, n) for c in b(n-1, k-1)]
def T(n, k): return add(b(n, k))
flatten([[T(n, k) for k in (0..n)] for n in (0..10)])
# Peter Luschny, Jul 29 2013

Crossrefs

Row sums give A226037.

Cf. A065567, A096179, A181854.

Sequence in context: A156367 A193593 A308616 * A008276 A094638 A196844

Adjacent sequences: A181850 A181851 A181852 * A181854 A181855 A181856

Keyword

nonn,tabl

Author

Peter Luschny, Dec 06 2010

Status

approved