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

0,5

Comments

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).

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

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

The 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

Maple

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

Crossrefs

Cf. A075362 (without column k=0), A025581, A025581.

Sequence in context: A112635 A222757 A004568 * A175630 A021420 A330473

Adjacent sequences: A079901 A079902 A079903 * A079905 A079906 A079907

Keyword

nonn,easy,tabl

Author

Reinhard Zumkeller, Feb 21 2003

Status

approved