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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}. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A156367 A193593 A308616 * A008276 A094638 A196844
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 06 2010
STATUS
approved

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Last modified April 22 01:34 EDT 2024. Contains 371887 sequences. (Running on oeis4.)