A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 7, 1, 1, 5, 26, 32, 12, 1, 1, 6, 57, 122, 92, 20, 1, 1, 7, 120, 423, 582, 252, 33, 1, 1, 8, 247, 1389, 3333, 2598, 681, 54, 1, 1, 9, 502, 4414, 18054, 24117, 11451, 1815, 88, 1, 1, 10, 1013, 13744, 94684, 210990, 172980, 49566, 4807, 143, 1

Offset: 0

Author

Gary W. Adamson, Aug 14 2005

Keywords

• easy
• nonn
• tabl

Comments

Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.

Row sums are given by A135934. - Emanuele Munarini, Dec 05 2017

Examples

			Triangle begins
  1....1....2....3....5....8...13....F(k+1)
  1
  1....1
  1....2....1
  1....3....4....1
  1....4...11....7....1
  1....5...26...32...12....1
  1....6...57..122...92...20....1
For example, T(6,3) = 122 = 26 + 3*32 = T(5,2) + F(4)*T(5,3).
		

Links

• Michel Marcus, Rows n = 0..20, flattened

Crossrefs

Cf. A111577, A111578, A111579, A008277, A039755, A135934.

Programs

Mathematica

(* To generate the triangle *)
Grid[RecurrenceTable[{F[n,k] == F[n-1,k-1] + Fibonacci[k+1] F[n-1,k], F[0,k] == KroneckerDelta[k]}, F, {n,0,10}, {k,0,10}]] (* Emanuele Munarini, Dec 05 2017 *)

PARI

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k-1) + fibonacci(k+1)*T(n-1, k))); \\ Michel Marcus, May 25 2024

Formula

T(n, k) = T(n-1, k-1) + F(k+1)*T(n-1, k) where F(n)=A000045(n).

Column k has g.f. x^k/Product_{j=0..k} (1 - F(j+1)*x).

Extensions

Edited by Paul Barry, Nov 14 2005