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A126988
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Triangle read by rows: T(n,k) = n/k if k is a divisor of n; T(n,k) = 0 if k is not a divisor of n (1 <= k <= n).
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60
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1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 0, 0, 0, 1, 6, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 9, 0, 3, 0, 0, 0, 0, 0, 1, 10, 5, 0, 0, 2, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1
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OFFSET
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1,2
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COMMENTS
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k-th column (k=0,1,2,...) is (1,2,3,...) interspersed with n consecutive zeros starting after the "1".
The nonzero entries of row n are the divisors of n in decreasing order. - Emeric Deutsch, Jan 17 2007
T(n,k) is the number of k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019
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REFERENCES
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David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, Inc, 2005, Appendix B.
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LINKS
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FORMULA
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G.f. of column k: z^k/(1-z^k)^2 (k=1,2,...).
G.f.: G(t,z) = Sum_{k>=1} t^k*z^k/(1-z^k)^2. (End)
G.f.: F(x,z) = log(1/(Product_{n >= 1} (1 - x*z^n))) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) = x*z + (2*x + x^2)*z^2/2 + (3*x + x^3)*z^3/3 + .... Note, exp(F(x,z)) is a g.f. for A008284 (with an additional term T(0,0) equal to 1). - Peter Bala, Jan 13 2015
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EXAMPLE
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First few rows of the triangle are:
1;
2, 1;
3, 0, 1;
4, 2, 0, 1;
5, 0, 0, 0, 1;
6, 3, 2, 0, 0, 1;
7, 0, 0, 0, 0, 0, 1;
8, 4, 0, 2, 0, 0, 0, 1;
9, 0, 3, 0, 0, 0, 0, 0, 1;
10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
...
sigma(12) = 28, from 12th row = (12 + 6 + 4 + 3 + 2 + 1), deleting the zeros, from left to right.
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of k's are [6, 3, 2, 0, 0, 1] respectively, equaling the 6th row of triangle. - Omar E. Pol, Nov 25 2019
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MAPLE
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A126988:=proc(n, k) if type(n/k, integer)=true then n/k else 0 fi end: for n from 1 to 12 do seq(A126988(n, k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 17 2007
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MATHEMATICA
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Table[If[Mod[n, m]==0, n/m, 0], {n, 1, 12}, {m, 1, n}]//Flatten (* Roger L. Bagula, Sep 06 2008, simplified by Franklin T. Adams-Watters, Aug 24 2011 *)
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PROG
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(Haskell)
a126988 n k = a126988_tabl !! (n-1) !! (k-1)
a126988_row n = a126988_tabl !! (n-1)
a126988_tabl = zipWith (zipWith (*)) a010766_tabl a051731_tabl
(PARI) {T(n, k) = if(n%k==0, n/k, 0)}; \\ G. C. Greubel, May 29 2019
(Magma) [[(n mod k) eq 0 select n/k else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, May 29 2019
(Sage)
def T(n, k):
if (n%k==0): return n/k
else: return 0
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 29 2019
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CROSSREFS
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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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