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%I #13 May 12 2015 20:08:41
%S 0,0,1,0,2,4,0,3,6,9,0,4,8,12,16,0,5,10,15,20,25,0,6,12,18,24,30,36,0,
%T 7,14,21,28,35,42,49,0,8,16,24,32,40,48,56,64,0,9,18,27,36,45,54,63,
%U 72,81,0,10,20,30,40,50,60,70,80,90,100,0,11,22,33,44,55,66,77,88,99,110,121
%N Triangle read by rows: T(n, k) = n*k, 0<=k<=n.
%C T(n, k) = if k=0 then 0 else T(n,k-1)+n;
%C T(n, 0)=1; T(n, 1)=n for n>0; T(n, 2)=A005843(n) for n>1; T(n, 3)=A008585(n) for n>2; T(n, 4)=A008586(n) for n>3;
%C T(n, n-2)=A005563(n-1) for n>1; T(n, n-1)=A002378(n-1) for n>0; T(n, n)=A000290(n).
%C See the comment in A025581 on a problem posed by François Viète (Vieta) 1593, where this triangle is related to A025581 and A257238. - _Wolfdieter Lang_, May 12 2015
%F T(n, k) = n*k, 0 <= k <= n.
%F T(n, k) = (A257238(n, k) - A025581(n, k)^3) / (3*A025581(n, k)). See the Viète comment above. - _Wolfdieter Lang_, May 12 2015
%F From _Robert Israel_, May 12 2015: (Start)
%F G.f. as triangle: (1 + x*y - 2*x^2*y)*x*y/((1-x)^2*(1-x*y)^3).
%F G.f. as sequence: -Sum(n >= 0, (n^2-n)*x^(n*(n+1)/2))/(1-x) + Sum(n >= 1, x^(n*(n+1)/2)) * x/(1-x)^2. These sums are related to Jacobi Theta functions.
%F (End)
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 0
%e 1: 0 1
%e 2: 0 2 4
%e 3: 0 3 6 9
%e 4: 0 4 8 12 16
%e 5: 0 5 10 15 20 25
%e 6: 0 6 12 18 24 30 36
%e 7: 0 7 14 21 28 35 42 49
%e 8: 0 8 16 24 32 40 48 56 64
%e 9: 0 9 18 27 36 45 54 63 72 81
%e 10: 0 10 20 30 40 50 60 70 80 90 100
%e ... - _Wolfdieter Lang_, May 12 2015
%p seq(seq(n*k,k=0..n),n=0..10); # _Robert Israel_, May 12 2015
%Y Cf. A075362 (without column k=0), A025581, A025581.
%K nonn,easy,tabl
%O 0,5
%A _Reinhard Zumkeller_, Feb 21 2003
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