%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)): 1, 3, 4, 7, 6, 12, 8, 15,... sigma(n) is the sum of the divisors of the integer n. The sequence of parsed terms in sigma(n) is the reversal of nonzero row terms in the present triangle.
%C kth 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/(1z^k)^2 (k=1,2,...). G.f.: G(t,z)=Sum(t^k*z^k/(1z^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) = 28 = (from tables): (1 + 2 + 3 + 4 + 6 + 12).
%e sigma(12) = 28, from 12th row of A126988 = (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. AdamsWatters_, Aug 24 2011 *)
%o (Haskell)
%o a126988 n k = a126988_tabl !! (n1) !! (k1)
%o a126988_row n = a126988_tabl !! (n1)
%o a126988_tabl = zipWith (zipWith (*)) a010766_tabl a051731_tabl
%o  _Reinhard Zumkeller_, Jan 20 2014
%Y Cf. A000203, A008284.
%K nonn,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. AdamsWatters_, Aug 24 2011
