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%I #39 Sep 05 2023 21:23:57
%S 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,
%T 494,282,60,1,28,288,1380,3245,3740,2034,420,1,36,456,2716,8293,13268,
%U 11338,4908,840,1,45,726,5578,22207,47351,57598,40602,15564,2520
%N 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}.
%C The C(n,k) are also called combinations of n with size k (see A181842).
%C Main diagonal gives: A003418. Lower diagonal gives: A094308. Column k=1 gives: A000217. - _Alois P. Heinz_, Jul 29 2013
%H Alois P. Heinz, <a href="/A181853/b181853.txt">Rows n = 0..46, flattened</a>
%e [0] 1
%e [1] 1 1
%e [2] 1 3 2
%e [3] 1 6 11 6
%e [4] 1 10 31 34 12
%e [5] 1 15 81 189 182 60
%e [6] 1 21 141 393 494 282 60
%p with(combstruct):
%p a181853_row := proc(n) local k,L,l,R,comb;
%p R := NULL;
%p for k from 0 to n do
%p L := 0;
%p comb := iterstructs(Combination(n),size=k):
%p while not finished(comb) do
%p l := nextstruct(comb);
%p L := L + ilcm(op(l));
%p od;
%p R := R,L;
%p od;
%p R end:
%p # second Maple program:
%p b:= proc(n, k) option remember; `if`(k=0, [1],
%p [`if`(k<n, b(n-1, k), [])[], seq(ilcm(c, n), c=b(n-1, k-1))])
%p end:
%p T:= (n, k)-> add(c, c=b(n, k)):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Jul 29 2013
%p # third Maple program:
%p b:= proc(n, m) option remember; expand(`if`(n=0, m,
%p b(n-1, ilcm(m, n))*x+b(n-1, m)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Sep 05 2023
%t 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 *)
%o (Sage) # (After Alois P. Heinz)
%o @CachedFunction
%o def b(n, k):
%o if k == 0: return [1]
%o w = b(n-1, k) if k<n else [0]
%o return w + [lcm(c,n) for c in b(n-1, k-1)]
%o def T(n, k): return add(b(n, k))
%o flatten([[T(n, k) for k in (0..n)] for n in (0..10)])
%o # _Peter Luschny_, Jul 29 2013
%Y Row sums give A226037.
%Y Cf. A065567, A096179, A181854.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Dec 06 2010
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