OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Column k has e.g.f.: (x^k/k!)*(cos(x) + sin(x)).
T(n, k) = binomial(n,k)*(cos(Pi*(n-k)/2) + sin(Pi*(n-k)/2)).
Sum_{k=0..n} T(n, k) = A009545(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117441(n) (upward diagonal sums).
G.f.: (1 + x - x*y)/(1 - 2*x*y + x^2*(1 + y^2)). - Stefano Spezia, Mar 10 2024
EXAMPLE
Triangle begins:
1;
1, 1;
-1, 2, 1;
-1, -3, 3, 1;
1, -4, -6, 4, 1;
1, 5, -10, -10, 5, 1;
-1, 6, 15, -20, -15, 6, 1;
-1, -7, 21, 35, -35, -21, 7, 1;
MATHEMATICA
Table[Binomial[n, k]*(Cos[Pi*(n-k)/2] +Sin[Pi*(n-k)/2]), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
PROG
(Sage) flatten([[binomial(n, k)*( cos(pi*(n-k)/2) + sin(pi*(n-k)/2) ) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 16 2006
STATUS
approved