This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334997 #41 Mar 31 2024 17:26:28
%S A334997 1,1,1,1,2,1,1,2,3,1,1,3,3,4,1,1,2,6,4,5,1,1,4,3,10,5,6,1,1,2,9,4,15,
%T A334997 6,7,1,1,4,3,16,5,21,7,8,1,1,3,10,4,25,6,28,8,9,1,1,4,6,20,5,36,7,36,
%U A334997 9,10,1,1,2,9,10,35,6,49,8,45,10,11,1,1,6,3,16,15,56,7,64,9,55,11,12,1
%N A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.
%C A334997 T(n, k) is called the generalized divisor function (see Beekman).
%C A334997 As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - _Gus Wiseman_, Aug 04 2022
%D A334997 Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.
%H A334997 Stefano Spezia, <a href="/A334997/b334997.txt">First 150 antidiagonals of the array, flattened</a>
%H A334997 Richard Beekman, <a href="https://www.researchgate.net/publication/341090354_A_General_Solution_of_the_Ferris_Wheel_Problem">A General Solution of the Ferris Wheel Problem</a>.
%F A334997 T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
%F A334997 T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
%F A334997 T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).
%e A334997 From _Gus Wiseman_, Aug 04 2022: (Start)
%e A334997 Array begins:
%e A334997 k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
%e A334997 n=1: 1 1 1 1 1 1 1 1 1
%e A334997 n=2: 1 2 3 4 5 6 7 8 9
%e A334997 n=3: 1 2 3 4 5 6 7 8 9
%e A334997 n=4: 1 3 6 10 15 21 28 36 45
%e A334997 n=5: 1 2 3 4 5 6 7 8 9
%e A334997 n=6: 1 4 9 16 25 36 49 64 81
%e A334997 n=7: 1 2 3 4 5 6 7 8 9
%e A334997 n=8: 1 4 10 20 35 56 84 120 165
%e A334997 The T(4,5) = 21 chains:
%e A334997 (1,1,1,1,1) (4,2,1,1,1) (4,4,2,2,2)
%e A334997 (2,1,1,1,1) (4,2,2,1,1) (4,4,4,1,1)
%e A334997 (2,2,1,1,1) (4,2,2,2,1) (4,4,4,2,1)
%e A334997 (2,2,2,1,1) (4,2,2,2,2) (4,4,4,2,2)
%e A334997 (2,2,2,2,1) (4,4,1,1,1) (4,4,4,4,1)
%e A334997 (2,2,2,2,2) (4,4,2,1,1) (4,4,4,4,2)
%e A334997 (4,1,1,1,1) (4,4,2,2,1) (4,4,4,4,4)
%e A334997 The T(6,3) = 16 chains:
%e A334997 (1,1,1) (3,1,1) (6,2,1) (6,6,1)
%e A334997 (2,1,1) (3,3,1) (6,2,2) (6,6,2)
%e A334997 (2,2,1) (3,3,3) (6,3,1) (6,6,3)
%e A334997 (2,2,2) (6,1,1) (6,3,3) (6,6,6)
%e A334997 The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
%e A334997 1
%e A334997 1 1
%e A334997 1 2 1
%e A334997 1 2 3 1
%e A334997 1 3 3 4 1
%e A334997 1 2 6 4 5 1
%e A334997 1 4 3 10 5 6 1
%e A334997 1 2 9 4 15 6 7 1
%e A334997 1 4 3 16 5 21 7 8 1
%e A334997 1 3 10 4 25 6 28 8 9 1
%e A334997 1 4 6 20 5 36 7 36 9 10 1
%e A334997 1 2 9 10 35 6 49 8 45 10 11 1
%e A334997 (End)
%t A334997 T[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; Table[T[n-k,k],{n,1,13},{k,0,n-1}]//Flatten
%o A334997 (PARI) T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
%o A334997 matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ _Michel Marcus_, May 20 2020
%Y A334997 Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
%Y A334997 Column k = 2 of the array is A007425.
%Y A334997 Column k = 3 of the array is A007426.
%Y A334997 Column k = 4 of the array is A061200.
%Y A334997 The transpose of the array is A077592.
%Y A334997 The subdiagonal n = k + 1 of the array is A163767.
%Y A334997 The version counting all multisets of divisors (not just chains) is A343658.
%Y A334997 The strict case is A343662 (row sums: A337256).
%Y A334997 Diagonal n = k of the array is A343939.
%Y A334997 Antidiagonal sums of the array (or row sums of the triangle) are A343940.
%Y A334997 A067824(n) counts strict chains of divisors starting with n.
%Y A334997 A074206(n) counts strict chains of divisors from n to 1.
%Y A334997 A146291 counts divisors by Omega.
%Y A334997 A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
%Y A334997 A253249(n) counts nonempty chains of divisors of n.
%Y A334997 A334996(n,k) counts strict length k chains of divisors from n to 1.
%Y A334997 A337255(n,k) counts strict length k chains of divisors starting with n.
%Y A334997 Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.
%K A334997 nonn,tabl,mult
%O A334997 1,5
%A A334997 _Stefano Spezia_, May 19 2020
%E A334997 Duplicate term removed by _Stefano Spezia_, Jun 03 2020