A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

0, 0, 1, 0, 1, 3, 0, 1, 3, 7, 0, 1, 3, 8, 15, 0, 1, 3, 9, 21, 31, 0, 1, 3, 10, 27, 55, 63, 0, 1, 3, 11, 33, 81, 144, 127, 0, 1, 3, 12, 39, 109, 243, 377, 255, 0, 1, 3, 13, 45, 139, 360, 729, 987, 511, 0, 1, 3, 14, 51, 171, 495, 1189, 2187, 2584, 1023, 0, 1, 3, 15, 57, 205, 648

Offset: 0

Author

Paul Barry, May 06 2003

Keywords

• easy
• nonn
• tabl

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.

Examples

			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, ...
  ...
		

Links

• OEIS, Transformations of integer sequences.

Crossrefs

Rows include A000225 (n=0), A001906 (n=1), A000244 (n=2), A006190 (n=3), A007482 (n=4), A030195 (n=5), A015521 (n=6).

Cf. A083856, A083861 (binomial transform), A083862 (main diagonal).

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

Extensions

Various sections edited by Petros Hadjicostas, Dec 24 2019