A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

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

Author

Gus Wiseman, Aug 23 2020

Keywords

• nonn
• tabf

Comments

Row n > 1 appears to be row n! of A334996.

Examples

			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
		

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!.

Cf. A022559, A124010, A167865, A251683, A337070, A337074.

Programs

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!]}]