A079904 Triangle read by rows: T(n, k) = n*k, 0 <= k <= n.

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, 7, 14, 21, 28, 35, 42, 49, 0, 8, 16, 24, 32, 40, 48, 56, 64, 0, 9, 18, 27, 36, 45, 54, 63, 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

Offset: 0

Author

Reinhard Zumkeller, Feb 21 2003

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			Triangle T(n, k) begins:
  n\k 0  1  2  3  4  5  6  7  8  9  10 ...
  0:  0
  1:  0  1
  2:  0  2  4
  3:  0  3  6  9
  4:  0  4  8 12 16
  5:  0  5 10 15 20 25
  6:  0  6 12 18 24 30 36
  7:  0  7 14 21 28 35 42 49
  8:  0  8 16 24 32 40 48 56 64
  9:  0  9 18 27 36 45 54 63 72 81
  10: 0 10 20 30 40 50 60 70 80 90 100
  ... - _Wolfdieter Lang_, May 12 2015
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Cf. A005843, A008585, A008586, A005563, A002378, A000290.

Cf. A075362 (without column k=0), A002411 (row sums), A001105 (central terms).

Cf. A257238, A025581.

Programs

Maple

seq(seq(n*k, k=0..n), n=0..10); # Robert Israel, May 12 2015

Mathematica

Array[Range[0, #^2, #] &, 15, 0] (* Paolo Xausa, Mar 27 2025 *)

PARI

row(n) = vector(n+1, i, (i-1)*n); \\ Amiram Eldar, May 12 2025

Formula

T(n, k) = n*k, 0 <= k <= n.

T(n, k) = if k = 0 then 0 else T(n, k-1) + n.

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.

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).

T(n, k) = (A257238(n, k) - A025581(n, k)^3) / (3*A025581(n, k)). See the Viète comment above. - Wolfdieter Lang, May 12 2015

From Robert Israel, May 12 2015: (Start)

G.f. as triangle: (1 + x*y - 2*x^2*y)*x*y/((1 - x)^2*(1 - x*y)^3).

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)

T(n, k) = gcd(n, k) * lcm(n, k). - Peter Luschny, Mar 26 2025