A105060 Triangle read by rows in which the n-th row consists of the first n nonzero terms of A033312.

1, 1, 5, 1, 5, 23, 1, 5, 23, 119, 1, 5, 23, 119, 719, 1, 5, 23, 119, 719, 5039, 1, 5, 23, 119, 719, 5039, 40319, 1, 5, 23, 119, 719, 5039, 40319, 362879, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799

Offset

1,3

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

From G. C. Greubel, Mar 13 2023: (Start)

T(n, k) = T(n,k-1) + k*k!, with T(n, 1) = 1.

Sum_{k=1..n} T(n, k) = -A007489(n+2) + (n+4)*A007489(n+1) - (n+2)*A007489(n) - (n+1). (End)

Example

Triangle begins as:
  1;
  1, 5;
  1, 5, 23;
  1, 5, 23, 119;
  1, 5, 23, 119, 719;
  1, 5, 23, 119, 719, 5039;
  1, 5, 23, 119, 719, 5039, 40319;

Mathematica

a[n_]:= a[n]= If[n==1, 1, a[n-1] + k!*n];
Table[a[k], {n, 12}, {k, n}]//Flatten

Prog

(Magma)
function T(n, k)
  if k eq 1 then return 1;
  else return T(n, k-1) + k*Factorial(k);
  end if;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 13 2023
(SageMath)
@CachedFunction
def T(n, k):
    if (k==1): return 1
    else: return T(n, k-1) + k*factorial(k)
flatten([[T(n, k) for k in range(1, n+1)] for n in range(1, 10)]) # G. C. Greubel, Mar 13 2023

Crossrefs

Cf. A007489, A033312.

Sequence in context: A101692 A281105 A327283 * A229096 A290797 A200423

Adjacent sequences: A105057 A105058 A105059 * A105061 A105062 A105063

Keyword

nonn,easy,tabl

Author

Roger L. Bagula, Apr 05 2005

Status

approved