This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079904 #33 May 12 2025 08:29:59
%S A079904 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 A079904 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 A079904 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 A079904 Triangle read by rows: T(n, k) = n*k, 0 <= k <= n.
%C A079904 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
%H A079904 Paolo Xausa, <a href="/A079904/b079904.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%F A079904 T(n, k) = n*k, 0 <= k <= n.
%F A079904 T(n, k) = if k = 0 then 0 else T(n, k-1) + n.
%F A079904 T(n, 0) = 1. T(n, 1) = n for n > 0.
%F A079904 T(n, 2) = A005843(n) for n > 1.
%F A079904 T(n, 3) = A008585(n) for n > 2.
%F A079904 T(n, 4) = A008586(n) for n > 3.
%F A079904 T(n, n-2) = A005563(n-1) for n > 1.
%F A079904 T(n, n-1) = A002378(n-1) for n > 0.
%F A079904 T(n, n) = A000290(n).
%F A079904 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 A079904 From _Robert Israel_, May 12 2015: (Start)
%F A079904 G.f. as triangle: (1 + x*y - 2*x^2*y)*x*y/((1 - x)^2*(1 - x*y)^3).
%F A079904 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. (End)
%F A079904 T(n, k) = gcd(n, k) * lcm(n, k). - _Peter Luschny_, Mar 26 2025
%e A079904 Triangle T(n, k) begins:
%e A079904 n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e A079904 0: 0
%e A079904 1: 0 1
%e A079904 2: 0 2 4
%e A079904 3: 0 3 6 9
%e A079904 4: 0 4 8 12 16
%e A079904 5: 0 5 10 15 20 25
%e A079904 6: 0 6 12 18 24 30 36
%e A079904 7: 0 7 14 21 28 35 42 49
%e A079904 8: 0 8 16 24 32 40 48 56 64
%e A079904 9: 0 9 18 27 36 45 54 63 72 81
%e A079904 10: 0 10 20 30 40 50 60 70 80 90 100
%e A079904 ... - _Wolfdieter Lang_, May 12 2015
%p A079904 seq(seq(n*k, k=0..n), n=0..10); # _Robert Israel_, May 12 2015
%t A079904 Array[Range[0, #^2, #] &, 15, 0] (* _Paolo Xausa_, Mar 27 2025 *)
%o A079904 (PARI) row(n) = vector(n+1, i, (i-1)*n); \\ _Amiram Eldar_, May 12 2025
%Y A079904 Cf. A005843, A008585, A008586, A005563, A002378, A000290.
%Y A079904 Cf. A075362 (without column k=0), A002411 (row sums), A001105 (central terms).
%Y A079904 Cf. A257238, A025581.
%K A079904 nonn,easy,tabl
%O A079904 0,5
%A A079904 _Reinhard Zumkeller_, Feb 21 2003