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

Author

Roger L. Bagula, Apr 05 2005

Keywords

• nonn
• easy
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A007489, A033312.

Programs

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

Mathematica

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

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

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)