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%I #47 Aug 10 2022 02:41:44
%S 1,1,1,1,2,1,1,3,2,1,1,4,3,3,1,1,5,4,6,2,1,1,6,5,10,3,4,1,1,7,6,15,4,
%T 9,2,1,1,8,7,21,5,16,3,4,1,1,9,8,28,6,25,4,10,3,1,1,10,9,36,7,36,5,20,
%U 6,4,1,1,11,10,45,8,49,6,35,10,9,2,1,1,12,11,55,9,64,7,56,15,16,3,6,1
%N Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.
%C As an array with offset n=0, k=1, also the number of length n chains of divisors of k. - _Gus Wiseman_, Aug 04 2022
%H Alois P. Heinz, <a href="/A077592/b077592.txt">Antidiagonals n = 1..141, flattened</a>
%H Adolf Piltz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN271032898">Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst</a>, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Adolf_Piltz">Adolf Piltz</a>.
%F If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).
%F Columns are multiplicative.
%F Dirichlet g.f. for column k: Zeta(s)^k. - _Geoffrey Critzer_, Feb 16 2015
%F A(n,k) = A334997(k,n). - _Gus Wiseman_, Aug 04 2022
%e T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - _Geoffrey Critzer_, Feb 16 2015
%e From _Gus Wiseman_, May 03 2021: (Start)
%e Array begins:
%e k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
%e n=0: 1 1 1 1 1 1 1 1
%e n=1: 1 2 2 3 2 4 2 4
%e n=2: 1 3 3 6 3 9 3 10
%e n=3: 1 4 4 10 4 16 4 20
%e n=4: 1 5 5 15 5 25 5 35
%e n=5: 1 6 6 21 6 36 6 56
%e n=6: 1 7 7 28 7 49 7 84
%e n=7: 1 8 8 36 8 64 8 120
%e n=8: 1 9 9 45 9 81 9 165
%e The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 3 2 1
%e 1 4 3 3 1
%e 1 5 4 6 2 1
%e 1 6 5 10 3 4 1
%e 1 7 6 15 4 9 2 1
%e 1 8 7 21 5 16 3 4 1
%e 1 9 8 28 6 25 4 10 3 1
%e 1 10 9 36 7 36 5 20 6 4 1
%e 1 11 10 45 8 49 6 35 10 9 2 1
%e (End)
%p with(numtheory):
%p A:= proc(n,k) option remember; `if`(k=1, 1,
%p add(A(d, k-1), d=divisors(n)))
%p end:
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Feb 25 2015
%t tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* _Robert G. Wilson v_ *)
%t tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] & /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Sep 13 2020 *)
%t Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, _Gus Wiseman_, May 03 2021 *)
%t Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, _Gus Wiseman_, May 03 2021 *)
%Y Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
%Y Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
%Y Cf. A077593.
%Y Row n = 2 of the array is A007425.
%Y Row n = 3 of the array is A007426.
%Y Row n = 4 of the array is A061200.
%Y The diagonal n = k of the array (central column of the triangle) is A163767.
%Y The transpose of the array is A334997.
%Y Diagonal n = k of the array is A343939.
%Y Antidiagonal sums of the array (or row sums of the triangle) are A343940.
%Y A067824(n) counts strict chains of divisors starting with n.
%Y A074206(n) counts strict chains of divisors from n to 1.
%Y A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
%Y A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
%Y A253249(n) counts nonempty chains of divisors of n.
%Y A334996(n,k) counts strict length k chains of divisors from n to 1.
%Y A337255(n,k) counts strict length k chains of divisors starting with n.
%Y Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.
%K mult,nonn,tabl,look
%O 1,5
%A _Henry Bottomley_, Nov 08 2002
%E Typo in formula fixed by _Geoffrey Critzer_, Feb 16 2015
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