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A051731
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Triangle read by rows: T(n, k) = 1 if k divides n, T(n, k) = 0 otherwise, for 1 <= k <= n.
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267
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1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,1
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COMMENTS
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T(n, k) is the number of partitions of n into k equal parts. - Omar E. Pol, Apr 21 2018
This triangle is the lower triangular array L in the LU decomposition of the square array A003989. - Peter Bala, Oct 15 2023
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LINKS
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FORMULA
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{T(n, k)*k, k=1..n} setminus {0} = divisors(n).
Sum_{k=1..n} T(n, k)*k^i = sigma[i](n), where sigma[i](n) is the sum of the i-th power of the positive divisors of n.
Sum_{k=1..n} T(n, k)*k = A000203(n).
T(n, k) = T(n-k, k) for k <= n/2, T(n, k) = 0 for n/2 < k <= n-1, T(n, n) = 1.
Binomial transform (product by binomial matrix) is A101508.
Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). (End)
A054525 is the inverse of this triangle (as lower triangular matrix).
This triangle * [1, 2, 3, ...] = sigma(n) (A000203).
This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n. (End)
T(n, k) = 0^(n mod k).
From Mats Granvik, Jan 26 2010, Feb 10 2010, Feb 16 2010: (Start)
T(n, k) = Sum_{i=1..k-1} (T(n-i, k-1) - T(n-i, k)) for k > 1 and T(n, 1) = 1.
(Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula.) (End)
The determinant of this matrix where T(n, n) has been swapped with T(1,k) is equal to the n-th term of the Mobius function. - Mats Granvik, Jul 21 2012
T(n, k) = Sum_{y=1..n} Sum_{x=1..n} [GCD((x/y)*(k/n), n) = k]. - Mats Granvik, Dec 17 2023
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EXAMPLE
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The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
1: 1
2: 1 1
3: 1 0 1
4: 1 1 0 1
5: 1 0 0 0 1
6: 1 1 1 0 0 1
7: 1 0 0 0 0 0 1
8: 1 1 0 1 0 0 0 1
9: 1 0 1 0 0 0 0 0 1
10: 1 1 0 0 1 0 0 0 0 1
11: 1 0 0 0 0 0 0 0 0 0 1
12: 1 1 1 1 0 1 0 0 0 0 0 1
13: 1 0 0 0 0 0 0 0 0 0 0 0 1
14: 1 1 0 0 0 0 1 0 0 0 0 0 0 1
15: 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1
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MAPLE
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A051731 := proc(n, k) if n mod k = 0 then 1 else 0 end if end proc:
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MATHEMATICA
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Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]
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PROG
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(PARI)
(Haskell)
a051731 n k = 0 ^ mod n k
a051731_row n = a051731_tabl !! (n-1)
a051731_tabl = map (map a000007) a048158_tabl
(Sage)
A051731_row = lambda n: [int(k.divides(n)) for k in (1..n)]
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CROSSREFS
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KEYWORD
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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EXTENSIONS
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STATUS
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approved
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