Search: a163767 -id:a163767
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A304965
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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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+20
3
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).
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MAPLE
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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:
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A321287
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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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+20
1
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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
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OFFSET
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0,3
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A305049
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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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+20
0
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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
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).
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MAPLE
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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:
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A060690
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a(n) = binomial(2^n + n - 1, n).
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+10
32
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1, 2, 10, 120, 3876, 376992, 119877472, 131254487936, 509850594887712, 7145544812472168960, 364974894538906616240640, 68409601066028072105113098240, 47312269462735023248040155132636160, 121317088003402776955124829814219234385920
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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MAPLE
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with(combinat): for n from 0 to 20 do printf(`%d, `, binomial(2^n+n-1, n)) od:
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MATHEMATICA
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Table[Binomial[2^n+n-1, n], {n, 0, 20}] (* Harvey P. Dale, Apr 19 2012 *)
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PROG
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(PARI) a(n)=binomial(2^n+n-1, n)
(PARI) {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
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014
(Sage) [binomial(2^n +n-1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n-1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
(Python)
from math import comb
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
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EXTENSIONS
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STATUS
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approved
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A334997
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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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+10
25
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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)
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MATHEMATICA
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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
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PROG
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(PARI) 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
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CROSSREFS
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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.
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.
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.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A077592
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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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+10
23
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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)
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MAPLE
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with(numtheory):
A:= proc(n, k) option remember; `if`(k=1, 1,
add(A(d, k-1), d=divisors(n)))
end:
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
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.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A346148
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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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+10
8
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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
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OFFSET
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1,5
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 ...
...
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MATHEMATICA
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T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
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PROG
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(Python)
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))
(PARI) 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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A343662
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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.
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+10
7
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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
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OFFSET
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1,4
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[Reverse/@Subsets[Divisors[n], {k}], And@@Divisible@@@Partition[#, 2, 1]&]], {n, 15}, {k, 0, PrimeOmega[n]+1}]
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CROSSREFS
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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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A343935
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Number of ways to choose a multiset of n divisors of n.
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+10
4
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.
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EXAMPLE
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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}
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MATHEMATICA
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multchoo[n_, k_]:=Binomial[n+k-1, k];
Table[multchoo[DivisorSigma[0, n], n], {n, 25}]
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PROG
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(Python)
from math import comb
from sympy import divisor_count
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CROSSREFS
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Choosing n divisors of n - 1 gives A343936.
The version for chains of divisors is A343939.
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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A343939
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Number of n-chains of divisors of n.
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+10
4
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[Tuples[Divisors[n], n], OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#, 2, 1]&]], {n, 10}]
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CROSSREFS
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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.
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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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