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

0,5

Comments

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

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

Links

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

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).

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A063841 A256161 A137596 * A336573 A124834 A271465

Adjacent sequences: A111666 A111667 A111668 * A111670 A111671 A111672

Keyword

easy,nonn,tabl

Author

Gary W. Adamson, Aug 14 2005

Extensions

Edited by Paul Barry, Nov 14 2005

Status

approved