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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). 53

%I

%S 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,

%T 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,

%U 6,4,3,0,2,0,0,0,0,0,1

%N 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).

%C Row sums = A000203, sigma(n).

%C k-th column (k=0,1,2...) is (1,2,3,...) interspersed with n consecutive zeros starting after the "1".

%C The nonzero entries of row n are the divisors of n in decreasing order. - _Emeric Deutsch_, Jan 17 2007

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

%H Reinhard Zumkeller, <a href="/A126988/b126988.txt">Rows n = 1..125 of triangle, flattened</a>

%F G.f. of column k: z^k/(1-z^k)^2 (k=1,2,...). G.f.: G(t,z)=Sum(t^k*z^k/(1-z^k)^2,k=1..infinity). - _Emeric Deutsch_, Jan 17 2007

%F 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

%F T(n,k) = A010766(n,k)*A051731(n,k), k=1..n. - _Reinhard Zumkeller_, Jan 20 2014

%e First few rows of the triangle are:

%e 1;

%e 2, 1;

%e 3, 0, 1;

%e 4, 2, 0, 1;

%e 5, 0, 0, 0, 1;

%e 6, 3, 2, 0, 0, 1;

%e 7, 0, 0, 0, 0, 0, 1;

%e 8, 4, 0, 2, 0, 0, 0, 1;

%e 9, 0, 3, 0, 0, 0, 0, 0, 1;

%e 10, 5, 0, 0, 2, 0, 0, 0, 0, 1;

%e ...

%e sigma(12) = A000203(n) = 28.

%e sigma(12) = 28, from 12th row = (12 + 6 + 4 + 3 + 2 + 1), deleting the zeros, from left to right.

%p 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

%t t[n_, m_] = If[Mod[n, m] == 0, n/m, 0]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[%] (* _Roger L. Bagula_, Sep 06 2008, simplified by _Franklin T. Adams-Watters_, Aug 24 2011 *)

%o a126988 n k = a126988_tabl !! (n-1) !! (k-1)

%o a126988_row n = a126988_tabl !! (n-1)

%o a126988_tabl = zipWith (zipWith (*)) a010766_tabl a051731_tabl

%o -- _Reinhard Zumkeller_, Jan 20 2014

%Y Cf. A000203, A008284.

%K nonn,easy,tabl

%O 1,2

%A _Gary W. Adamson_, Dec 31 2006

%E Edited by _N. J. A. Sloane_, Jan 24 2007

%E Comment from _Emeric Deutsch_ made name by _Franklin T. Adams-Watters_, Aug 24 2011

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