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;
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
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
KEYWORD
AUTHOR
Reinhard Zumkeller, Feb 21 2003
STATUS
approved