A163767 -id:A163767 - cp's OEIS frontend

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A163767.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).

A321287 Expansion of Product_{k>=1} (1 + x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 2, 5, 14, 22, 70, 109, 318, 551, 1203, 2136, 5752, 9263, 20641, 37151, 85084, 144918, 317356, 546730, 1196302, 2076810, 4281584, 7459351, 15860805, 27146911, 54715933, 95712097, 194059563, 334322338, 663159101, 1147479053, 2270647257, 3923732160, 7587368893

Offset: 0

Seiichi Manyama, Nov 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A163767, A304965, A321192.

tau[n_,1] = 1; tau[n_,k_]:=tau[n,k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 40; CoefficientList[Series[Product[(1+x^k)^tau[k,k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 03 2018, after Robert G. Wilson v *)

A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

Original entry on oeis.org

1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Invert transform of A163767.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j$2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 24 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).

A060690 a(n) = binomial(2^n + n - 1, n).

Original entry on oeis.org

1, 2, 10, 120, 3876, 376992, 119877472, 131254487936, 509850594887712, 7145544812472168960, 364974894538906616240640, 68409601066028072105113098240, 47312269462735023248040155132636160, 121317088003402776955124829814219234385920

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001

nonn

Also the number of n X n (0,1) matrices modulo rows permutation (by symmetry this is the same as the number of (0,1) matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other. The total number of (0,1) matrices is in sequence A002416.

Row sums of A220886. - Geoffrey Critzer, Nov 20 2014

Harry J. Smith, Table of n, a(n) for n = 0..59

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), this sequence (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A002416, A060336, A088309, A163767.
Cf. A136555, A220886.
Main diagonal of A092056.
Central terms of A137153.

[Binomial(2^n +n-1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

with(combinat): for n from 0 to 20 do printf(`%d,`,binomial(2^n+n-1, n)) od:

Table[Binomial[2^n+n-1,n],{n,0,20}] (* Harvey P. Dale, Apr 19 2012 *)

PARI
```
a(n)=binomial(2^n+n-1,n)
```

{a(n)=polcoeff(sum(k=0,n,(-log(1-2^k*x +x*O(x^n)))^k/k!),n)} \\ Paul D. Hanna, Dec 29 2007

a(n) = sum(k=0, n, stirling(n,k,1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014

from math import comb
def A060690(n): return comb((1<Chai Wah Wu, Jul 05 2024

[binomial(2^n +n-1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = [x^n] 1/(1-x)^(2^n).
a(n) = (1/n!)*Sum_{k=0..n} ( (-1)^(n-k)*Stirling1(n, k)*2^(k*n) ). - Vladeta Jovovic, May 28 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2^n+n,k) - Vladeta Jovovic, Jan 21 2008
a(n) = Sum_{k=0..n} Stirling1(n,k)*(2^n+n-1)^k/n!. - Vladeta Jovovic, Jan 21 2008
G.f.: A(x) = Sum_{n>=0} [ -log(1 - 2^n*x)]^n / n!. More generally, Sum_{n>=0} [ -log(1 - q^n*x)]^n/n! = Sum_{n>=0} C(q^n+n-1,n)*x^n ; also Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n. - Paul D. Hanna, Dec 29 2007
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
a(n) = A163767(2^n). - Alois P. Heinz, Jun 12 2024

More terms from James Sellers, Apr 20 2001

Edited by N. J. A. Sloane, Mar 17 2008

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

Original entry on oeis.org

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, 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, 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

Offset: 1

Stefano Spezia, May 19 2020

T(n, k) is called the generalized divisor function (see Beekman).

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

			From _Gus Wiseman_, Aug 04 2022: (Start)
Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4   9  16  25  36  49  64  81
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
The T(4,5) = 21 chains:
  (1,1,1,1,1)  (4,2,1,1,1)  (4,4,2,2,2)
  (2,1,1,1,1)  (4,2,2,1,1)  (4,4,4,1,1)
  (2,2,1,1,1)  (4,2,2,2,1)  (4,4,4,2,1)
  (2,2,2,1,1)  (4,2,2,2,2)  (4,4,4,2,2)
  (2,2,2,2,1)  (4,4,1,1,1)  (4,4,4,4,1)
  (2,2,2,2,2)  (4,4,2,1,1)  (4,4,4,4,2)
  (4,1,1,1,1)  (4,4,2,2,1)  (4,4,4,4,4)
The T(6,3) = 16 chains:
  (1,1,1)  (3,1,1)  (6,2,1)  (6,6,1)
  (2,1,1)  (3,3,1)  (6,2,2)  (6,6,2)
  (2,2,1)  (3,3,3)  (6,3,1)  (6,6,3)
  (2,2,2)  (6,1,1)  (6,3,3)  (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
  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  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  9 10  1
  1  2  9 10 35  6 49  8 45 10 11  1
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 150 antidiagonals of the array, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

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.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
The strict case is A343662 (row sums: A337256).
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291 counts divisors by Omega.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.

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

T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020

T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
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).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).

Duplicate term removed by Stefano Spezia, Jun 03 2020

A077592 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.

Original entry on oeis.org

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, 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, 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

Offset: 1

Henry Bottomley, Nov 08 2002

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

			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
From _Gus Wiseman_, May 03 2021: (Start)
Array begins:
       k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=0:  1   1   1   1   1   1   1   1
  n=1:  1   2   2   3   2   4   2   4
  n=2:  1   3   3   6   3   9   3  10
  n=3:  1   4   4  10   4  16   4  20
  n=4:  1   5   5  15   5  25   5  35
  n=5:  1   6   6  21   6  36   6  56
  n=6:  1   7   7  28   7  49   7  84
  n=7:  1   8   8  36   8  64   8 120
  n=8:  1   9   9  45   9  81   9 165
The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
  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  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  6  4  1
  1 11 10 45  8 49  6 35 10  9  2  1
(End)

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, 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.
Wikipedia, Adolf Piltz.

Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
Cf. A077593.
Row n = 2 of the array is A007425.
Row n = 3 of the array is A007426.
Row n = 4 of the array is A061200.
The diagonal n = k of the array (central column of the triangle) is A163767.
The transpose of the array is A334997.
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.

with(numtheory):
A:= proc(n,k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=divisors(n)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Feb 25 2015

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 *)
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 *)
Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, Gus Wiseman, May 03 2021 *)
Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, Gus Wiseman, May 03 2021 *)

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).
Columns are multiplicative.
Dirichlet g.f. for column k: Zeta(s)^k. - Geoffrey Critzer, Feb 16 2015
A(n,k) = A334997(k,n). - Gus Wiseman, Aug 04 2022

Typo in formula fixed by Geoffrey Critzer, Feb 16 2015

A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -2, -3, 1, -1, 1, -3, -4, 1, 1, -2, 3, -4, -5, 1, -1, 4, -3, 6, -5, -6, 1, 0, -2, 9, -4, 10, -6, -7, 1, 0, 0, -3, 16, -5, 15, -7, -8, 1, 1, 1, -1, -4, 25, -6, 21, -8, -9, 1, -1, 4, 3, -4, -5, 36, -7, 28, -9, -10, 1, 0, -2, 9, 6, -10, -6

Offset: 1

Sebastian Karlsson, Aug 20 2021

			  n\k| 1    2    3    4    5    6    7    8    9   10   11    12 ...
  ---+--------------------------------------------------------------
   1 | 1   -1   -1    0   -1    1   -1    0    0    1   -1     0 ...
   2 | 1   -2   -2    1   -2    4   -2    0    1    4   -2    -2 ...
   3 | 1   -3   -3    3   -3    9   -3   -1    3    9   -3    -9 ...
   4 | 1   -4   -4    6   -4   16   -4   -4    6   16   -4   -24 ...
   5 | 1   -5   -5   10   -5   25   -5  -10   10   25   -5   -50 ...
   6 | 1   -6   -6   15   -6   36   -6  -20   15   36   -6   -90 ...
   7 | 1   -7   -7   21   -7   49   -7  -35   21   49   -7  -147 ...
   8 | 1   -8   -8   28   -8   64   -8  -56   28   64   -8  -224 ...
   9 | 1   -9   -9   36   -9   81   -9  -84   36   81   -9  -324 ...
  10 | 1  -10  -10   45  -10  100  -10 -120   45  100  -10  -450 ...
  11 | 1  -11  -11   55  -11  121  -11 -165   55  121  -11  -605 ...
  12 | 1  -12  -12   66  -12  144  -12 -220   66  144  -12  -792 ...
  13 | 1  -13  -13   78  -13  169  -13 -286   78  169  -13 -1014 ...
  14 | 1  -14  -14   91  -14  196  -14 -364   91  196  -14 -1274 ...
  15 | 1  -15  -15  105  -15  225  -15 -455  105  225  -15 -1575 ...
  ...

Sebastian Karlsson, Antidiagonals n = 1..140, flattened

Row n=1..10 give A008683, A007427, A007428, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
Main diagonal gives A341837.
Cf. A077592, A163767,

T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2021 *)

T(n, k) = my(f=factor(k)); for (k=1, #f~, f[k,1] = binomial(n, f[k,2])*(-1)^f[k,2]; f[k,2]=1); factorback(f); \\ Michel Marcus, Aug 21 2021

from sympy import binomial, primefactors as pf, multiplicity as mult
from math import prod
def T(n, k):
    return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))

If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
Dirichlet g.f. of the n-th row: 1/zeta^n(s).
T(n, p) = -n.
T(n, n) = A341837(n).

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A343935 Number of ways to choose a multiset of n divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 84, 8, 165, 55, 286, 12, 6188, 14, 680, 816, 4845, 18, 33649, 20, 53130, 2024, 2300, 24, 2629575, 351, 3654, 4060, 237336, 30, 10295472, 32, 435897, 7140, 7770, 8436, 177232627, 38, 10660, 11480, 62891499, 42, 85900584, 44, 1906884, 2118760

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 multisets:
  {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}
       {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}
       {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}
              {3,3,3}  {1,1,2,2}  {1,1,5,5,5}
                       {1,1,2,4}  {1,5,5,5,5}
                       {1,1,4,4}  {5,5,5,5,5}
                       {1,2,2,2}
                       {1,2,2,4}
                       {1,2,4,4}
                       {1,4,4,4}
                       {2,2,2,2}
                       {2,2,2,4}
                       {2,2,4,4}
                       {2,4,4,4}
                       {4,4,4,4}

Diagonal n = k of A343658.
Choosing n divisors of n - 1 gives A343936.
The version for chains of divisors is A343939.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343661.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n],{n,25}]

from math import comb
from sympy import divisor_count
def A343935(n): return comb(divisor_count(n)+n-1,n) # Chai Wah Wu, Jul 05 2024

a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.

A343939 Number of n-chains of divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 chains:
  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)
       (2/1)  (3/1/1)  (2/1/1/1)  (5/1/1/1/1)
       (2/2)  (3/3/1)  (2/2/1/1)  (5/5/1/1/1)
              (3/3/3)  (2/2/2/1)  (5/5/5/1/1)
                       (2/2/2/2)  (5/5/5/5/1)
                       (4/1/1/1)  (5/5/5/5/5)
                       (4/2/1/1)
                       (4/2/2/1)
                       (4/2/2/2)
                       (4/4/1/1)
                       (4/4/2/1)
                       (4/4/2/2)
                       (4/4/4/1)
                       (4/4/4/2)
                       (4/4/4/4)

Diagonal n = k - 1 of the array A077592.
Chains of length n - 1 are counted by A163767.
Diagonal n = k of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005(n) counts divisors of n.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums: A337256).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

cp's OEIS Frontend

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A321287 Expansion of Product_{k>=1} (1 + x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

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A060690 a(n) = binomial(2^n + n - 1, n).

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Magma

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A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

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A077592 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.

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A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).

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