A058294 Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 7, 10, 7, 3, 1, 1, 4, 13, 30, 43, 30, 13, 4, 1, 1, 5, 21, 68, 157, 225, 157, 68, 21, 5, 1, 1, 6, 31, 130, 421, 972, 1393, 972, 421, 130, 31, 6, 1, 1, 7, 43, 222, 931, 3015, 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1

Offset

1,6

Comments

From Reinhard Zumkeller, Sep 14 2014: (Start)

T(n,k) = A102473(n,k), k=1..n;

T(n,k) = A102472(n,k-n+1), k=n..2*n-1;

T(n,n) = A001040(n). (End)

Links

Reinhard Zumkeller, >Rows n = 1..125 of table, flattened

Russell Walsmith, CL-Chemy Transforms Fibonacci-Type Sequences to Arrays

Formula

The j-th column S(j) is generated by a(n+1) = (n+j)*a(n) + a(n-1), a(0)=0, a(1)=1.

Example

Triangle begins:
  1;
  1, 1, 1;
  1, 2, 3,  2, 1;
  1, 3, 7, 10, 7, 3, 1;
  ...

Mathematica

t[n_, n_] = 1; t[n_, k_] := t[n, k] = If[n<k, 0, (n-1)*t[n-1, k] + t[n-2, k]]; row[n_] := With[{ro = Table[t[n, k], {k, 1, n}]}, Join[Reverse[ro], Rest[ro]]]; Array[row, 8] // Flatten (* Jean-François Alcover, Oct 05 2016 *)

Prog

(Haskell)
a058294 n k = a058294_tabf !! (n-1) !! (k-1)
a058294_row n = a058294_tabf !! (n-1)
a058294_tabf = [1] : zipWith (++) xss (map (tail . reverse) xss)
               where xss = tail a102473_tabl
-- Reinhard Zumkeller, Sep 14 2014

Crossrefs

A001040, A001053, A058307, A058308, A058309 are columns of this triangle.

Cf. A102472, A102473.

Sequence in context: A026323 A017838 A181567 * A323834 A082868 A219539

Adjacent sequences: A058291 A058292 A058293 * A058295 A058296 A058297

Keyword

nonn,tabf,nice,easy

Author

Russell Walsmith, Dec 07 2000

Status

approved