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A337107
Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.
8
1, 0, 1, 0, 1, 2, 0, 1, 6, 9, 4, 0, 1, 14, 45, 52, 20, 0, 1, 28, 183, 496, 655, 420, 105, 0, 1, 58, 633, 2716, 5755, 6450, 3675, 840, 0, 1, 94, 1659, 11996, 46235, 106806, 155869, 145384, 84276, 27720, 3960
OFFSET
1,6
COMMENTS
Row n > 1 appears to be row n! of A334996.
EXAMPLE
Triangle begins:
1
0 1
0 1 2
0 1 6 9 4
0 1 14 45 52 20
0 1 28 183 496 655 420 105
0 1 58 633 2716 5755 6450 3675 840
Row n = 4 counts the following chains:
24/1 24/2/1 24/4/2/1 24/8/4/2/1
24/3/1 24/6/2/1 24/12/4/2/1
24/4/1 24/6/3/1 24/12/6/2/1
24/6/1 24/8/2/1 24/12/6/3/1
24/8/1 24/8/4/1
24/12/1 24/12/2/1
24/12/3/1
24/12/4/1
24/12/6/1
MAPLE
b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
add(b(d), d=numtheory[divisors](n) minus {n})))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
seq(T(n), n=1..10); # Alois P. Heinz, Aug 23 2020
MATHEMATICA
nv=5;
chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#, n]&/@chnsc[d], {d, DeleteCases[Divisors[n], n]}], {n}], MemberQ[#, 1]&];
Table[Length[Select[chnsc[n!], Length[#]==k&]], {n, nv}, {k, 1+PrimeOmega[n!]}]
CROSSREFS
A097805 is the restriction to powers of 2.
A325617 is the maximal case.
A337105 gives row sums.
A337106 is column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A074206 counts chains of divisors from n to 1.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains starting with n!.
Sequence in context: A353451 A226573 A260693 * A362588 A367073 A176129
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Aug 23 2020
STATUS
approved