OFFSET
0,6
COMMENTS
Row n >= 0 of the array gives the solution to the recurrence b(k) = 3*b(k-1) + (n-2) * a(k-2) for k >= 2 with a(0) = 0 and a(1) = 1. These are the binomial transforms of the rows of the generalized Fibonacci numbers A083856.
FORMULA
T(n, k) = ((3 + sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) - ((3 - sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) for n, k >= 0.
O.g.f. of row n >= 0: -x/(-1 + 3*x + (n-2)*x^2) . - R. J. Mathar, Nov 23 2007
T(n,k) = Sum_{i = 0..k} binomial(k,i)*A083856(n,i). - Petros Hadjicostas, Dec 24 2019
EXAMPLE
Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
0, 1, 3, 7, 15, 31, 63, 127, 255, ...
0, 1, 3, 8, 21, 55, 144, 377, 987, ...
0, 1, 3, 9, 27, 81, 243, 729, 2187, ...
0, 1, 3, 10, 33, 109, 360, 1189, 3927, ...
0, 1, 3, 11, 39, 139, 495, 1763, 6279, ...
0, 1, 3, 12, 45, 171, 648, 2457, 9315, ...
...
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 06 2003
EXTENSIONS
Various sections edited by Petros Hadjicostas, Dec 24 2019
STATUS
approved