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A126988 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). 60
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums = A000203, sigma(n).

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

Alternating row sums give A000593. - Omar E. Pol, Feb 11 2018

T(n,k) is the number of k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

REFERENCES

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, Inc, 2005, Appendix B.

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

FORMULA

From Emeric Deutsch, Jan 17 2007: (Start)

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

T(n,k) = A010766(n,k)*A051731(n,k), k=1..n. - Reinhard Zumkeller, Jan 20 2014

EXAMPLE

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) = A000203(n) = 28.

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

-- Reinhard Zumkeller, Jan 20 2014

(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

CROSSREFS

Cf. A000203, A008284, A127446, A244051, A328361.

Sequence in context: A309229 A143239 A158951 * A280499 A130026 A113287

Adjacent sequences:  A126985 A126986 A126987 * A126989 A126990 A126991

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Dec 31 2006

EXTENSIONS

Edited by N. J. A. Sloane, Jan 24 2007

Comment from Emeric Deutsch made name by Franklin T. Adams-Watters, Aug 24 2011

STATUS

approved

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Last modified July 2 11:54 EDT 2020. Contains 335398 sequences. (Running on oeis4.)