%I A003991
%S A003991 1,2,2,3,4,3,4,6,6,4,5,8,9,8,5,6,10,12,12,10,6,7,12,15,16,15,12,7,8,
%T A003991 14,18,20,20,18,14,8,9,16,21,24,25,24,21,16,9,10,18,24,28,30,30,28,24,
%U A003991 18,10,11,20,27,32,35,36,35,32,27,20,11,12,22,30,36,40,42,42,40,36,30
%N A003991 Multiplication table read by antidiagonals: T(i,j) = ij, i>=1, j>=1.
%C A003991 Or, triangle read by rows, in which row n gives the numbers n*1, (n-1)*2, 
               (n-2)*3, ..., 2*(n-1), 1*n.
%C A003991 Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, 
               b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen (fvlamoen(AT)hotmail.com), 
               Aug 16 2001
%C A003991 A permutation of A061017. - Matthew Vandermast (ghodges14(AT)comcast.net), 
               Feb 28 2003
%C A003991 In the proof of countability of rational numbers they are arranged in 
               a square array. a(n) = p*q where p/q is the corresponding rational 
               number as read from the array. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               May 29 2003
%C A003991 Permanent of upper right n X n corner is A000442. - Marc LeBrun (mlb(AT)well.com), 
               Dec 11 2003
%C A003991 Row 12 gives total number of partridges, turtle doves, ... and drummers 
               drumming that you have received at the end of the Twelve Days of 
               Christmas song. - Alonso Del Arte, Jun 17 2005
%C A003991 Generated by additive equivalent of binomial theorem : T(i,j) = t(i)-t(j)-t(i-j), 
               where t(k)=k(k+1)/2 - Jon Perry (perry(AT)globalnet.co.uk), Nov 23 
               2005
%D A003991 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 
               1996, p. 46.
%H A003991 T. D. Noe, <a href="b003991.txt">Table of n, a(n) for n=1..5050</a>
%H A003991 A. Necer, <a href="http://www.emis.de/journals/JTNB/1997-2/jtnb9-2_english.html#jourelec">
               Series formelles et produit de Hadamard</a>
%F A003991 T(n, m)=m*(n-m+1).
%F A003991 Sum i=1..n Sum j=1..n a(n) = A000537(n) [Sum of first n cubes; or n-th 
               triangular number squared.] Determinant of all n X n contiguous subarrays 
               of A003991 is 0. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Sep 26 2004
%F A003991 G.f.: x * y / [ (1-x)^2 * (1-y)^2 ].
%F A003991 a(n)=(i-j+1)*j, where i=floor((1+sqr(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 08 2007
%F A003991 As an infinite lower triangular matrix equals A000012 * A002260; where 
               A000012 = (1; 1,1; 1,1,1;...) and A002260 = (1; 1,2; 1,2,3;...). 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 23 2007
%F A003991 t(n,m)=n/(1/m + 1/(n - m)) [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Feb 02 2009]
%e A003991 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 
               2009: (Start)
%e A003991 {1},
%e A003991 {2, 2},
%e A003991 {3, 4, 3},
%e A003991 {4, 6, 6, 4},
%e A003991 {5, 8, 9, 8, 5},
%e A003991 {6, 10, 12, 12, 10, 6},
%e A003991 {7, 12, 15, 16, 15, 12, 7},
%e A003991 {8, 14, 18, 20, 20, 18, 14, 8},
%e A003991 {9, 16, 21, 24, 25, 24, 21, 16, 9},
%e A003991 {10, 18, 24, 28, 30, 30, 28, 24, 18, 10},
%e A003991 {11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11},
%e A003991 {12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12},
%e A003991 {13, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13} (End)
%t A003991 Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson 
               v (rgwv(AT)rgwv.com), Oct 06 2007 *)
%t A003991 Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 
               2009: (Start)
%t A003991 Clear[f]; f[n_, m_] = n/(1/m + 1/(n - m));
%t A003991 Table[Table[f[n, m], {m, 1, n - 1}], {n, 2, 14}];
%t A003991 Flatten[%] (End)
%o A003991 (PARI) T(n,k) = if(k<1|k>n,0,k*(n+1-k))
%Y A003991 Main diagonal gives squares A000290. Anti-diagonal sums are tetrahedral 
               numbers A000292. See A004247 for another version.
%Y A003991 Cf. A003989, A003990, A003056, A049581, A000442, A027424.
%Y A003991 Cf. A002260.
%Y A003991 Sequence in context: A162619 A032355 A091257 this_sequence A131923 A119457 
               A065157
%Y A003991 Adjacent sequences: A003988 A003989 A003990 this_sequence A003992 A003993 
               A003994
%K A003991 tabl,nonn,nice,easy
%O A003991 1,2
%A A003991 Marc LeBrun (mlb(AT)well.com)
%E A003991 More terms from Michael Somos