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%I #24 Sep 13 2020 13:04:54
%S 1,9,9,45,9,81,9,165,45,81,9,405,9,81,81,495,9,405,9,405,81,81,9,1485,
%T 45,81,165,405,9,729,9,1287,81,81,81,2025,9,81,81,1485,9,729,9,405,
%U 405,81,9,4455,45,405,81,405,9,1485,81,1485,81,81,9,3645,9,81,405,3003,81
%N d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).
%H Seiichi Manyama, <a href="/A111219/b111219.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique Pérez Herrero)
%F G.f.: Sum_{k>=1} tau_8(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Oct 30 2018
%F Multiplicative with a(p^e) = binomial(e+8,8). - _Amiram Eldar_, Sep 13 2020
%t tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 9], {n, 65}] (* _Robert G. Wilson v_, Nov 02 2005 *)
%t tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 9], {n, 1, 100}] (* _Amiram Eldar_, Sep 13 2020 *)
%o (PARI) for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,x,numdiv(x)))))))),","))
%o (PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+8, 8)) \\ _Charles R Greathouse IV_, Oct 28 2017
%Y Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.
%Y Column k=9 of A077592.
%K mult,nonn
%O 1,2
%A _Gerald McGarvey_, Oct 25 2005
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