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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.

%I A337107 #11 Aug 24 2020 01:02:49
%S A337107 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,
%T A337107 58,633,2716,5755,6450,3675,840,0,1,94,1659,11996,46235,106806,155869,
%U A337107 145384,84276,27720,3960
%N A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.
%C A337107 Row n > 1 appears to be row n! of A334996.
%e A337107 Triangle begins:
%e A337107     1
%e A337107     0    1
%e A337107     0    1    2
%e A337107     0    1    6    9    4
%e A337107     0    1   14   45   52   20
%e A337107     0    1   28  183  496  655  420  105
%e A337107     0    1   58  633 2716 5755 6450 3675  840
%e A337107 Row n = 4 counts the following chains:
%e A337107   24/1  24/2/1   24/4/2/1   24/8/4/2/1
%e A337107         24/3/1   24/6/2/1   24/12/4/2/1
%e A337107         24/4/1   24/6/3/1   24/12/6/2/1
%e A337107         24/6/1   24/8/2/1   24/12/6/3/1
%e A337107         24/8/1   24/8/4/1
%e A337107         24/12/1  24/12/2/1
%e A337107                  24/12/3/1
%e A337107                  24/12/4/1
%e A337107                  24/12/6/1
%p A337107 b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
%p A337107       add(b(d), d=numtheory[divisors](n) minus {n})))
%p A337107     end:
%p A337107 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
%p A337107 seq(T(n), n=1..10);  # _Alois P. Heinz_, Aug 23 2020
%t A337107 nv=5;
%t A337107 chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],n]}],{n}],MemberQ[#,1]&];
%t A337107 Table[Length[Select[chnsc[n!],Length[#]==k&]],{n,nv},{k,1+PrimeOmega[n!]}]
%Y A337107 A097805 is the restriction to powers of 2.
%Y A337107 A325617 is the maximal case.
%Y A337107 A337105 gives row sums.
%Y A337107 A337106 is column k = 3.
%Y A337107 A000005 counts divisors.
%Y A337107 A000142 lists factorial numbers.
%Y A337107 A001055 counts factorizations.
%Y A337107 A074206 counts chains of divisors from n to 1.
%Y A337107 A027423 counts divisors of factorial numbers.
%Y A337107 A067824 counts chains of divisors starting with n.
%Y A337107 A076716 counts factorizations of factorial numbers.
%Y A337107 A253249 counts chains of divisors.
%Y A337107 A337071 counts chains starting with n!.
%Y A337107 Cf. A022559, A124010, A167865, A251683, A337070, A337074.
%K A337107 nonn,tabf
%O A337107 1,6
%A A337107 _Gus Wiseman_, Aug 23 2020