A136680 Triangle T(n, k) = f(k) for k < n+1, otherwise 0, where f(k) = f(k-1) + k^(k-2)*f(k-2) with f(0) = 0 and f(1) = 1, read by rows.

1, 1, 1, 1, 1, 4, 1, 1, 4, 20, 1, 1, 4, 20, 520, 1, 1, 4, 20, 520, 26440, 1, 1, 4, 20, 520, 26440, 8766080, 1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 41934828744960, 1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 41934828744960, 694027278828744960

Offset

1,6

Links

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Formula

T(k) = T(k-1) + n^(k-2)*T(k-2), with T(0) = 0, T(1) = 1.

T(n, k) = f(k) for k < n+1, otherwise 0, where f(k) = f(k-1) + k^(k-2)*f(k-2) with f(0) = 0 and f(1) = 1. - G. C. Greubel, Dec 01 2022

Example

Triangle begins as:
  1;
  1, 1;
  1, 1, 4;
  1, 1, 4, 20;
  1, 1, 4, 20, 520;
  1, 1, 4, 20, 520, 26440;
  1, 1, 4, 20, 520, 26440, 8766080;
  1, 1, 4, 20, 520, 26440, 8766080, 6939853440;
  1, 1, 4, 20, 520, 26440, 8766080, 6939853440, 41934828744960;

Mathematica

T[k_]:= T[k]= If[k<2, k, T[k-1] + n^(k-2)*T[k-2]];
Table[T[k], {n, 10}, {k, n}]//Flatten

Prog

(Magma)
function f(k)
  if k lt 2 then return k;
  else return f(k-1) + k^(k-2)*f(k-2);
  end if; return f;
end function;
A136680:= func< n, k | k le n select f(k) else 0 >;
[A136680(n, k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 01 2022
(SageMath)
@CachedFunction
def f(k):
    if (k<2): return k
    else: return f(k-1) + k^(k-2)*f(k-2)
def A136680(n, k): return f(k) if (k < n+1) else 0
flatten([[A136680(n, k) for k in range(1, n+1)] for n in range(1, 15)]) # G. C. Greubel, Dec 01 2022

Crossrefs

q-Fibonacci numbers include: A015459, A015460, A015461, A015462, A015463, A015464, A015465, A015467, A015468, A015469, A015470, A015473, A015474, A015475, A015476, A015477, A015479, A015480, A015481, A015482, A015484, A015485.

Sequence in context: A122185 A350000 A193752 * A288580 A035589 A021247

Adjacent sequences: A136677 A136678 A136679 * A136681 A136682 A136683

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 06 2008

Extensions

Edited by G. C. Greubel, Dec 01 2022

Status

approved