A105064 Triangle, read by rows, T(n, k) = T(n, k-1) + (k+1)*n!, T(n, 0) = 1.

1, 1, 3, 1, 5, 11, 1, 13, 31, 55, 1, 49, 121, 217, 337, 1, 241, 601, 1081, 1681, 2401, 1, 1441, 3601, 6481, 10081, 14401, 19441, 1, 10081, 25201, 45361, 70561, 100801, 136081, 176401, 1, 80641, 201601, 362881, 564481, 806401, 1088641, 1411201, 1774081

Offset

0,3

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

From G. C. Greubel, Mar 12 2023: (Start)

T(n, k) = T(n, k-1) + (k+1)*n!, T(n, 0) = 1.

T(n, n-1) = (1/2)*((n^2 + n - 2)*n! + 2).

T(n, n) = (1/2)*(n*(n+3)*n! + 2). (End)

Example

Triangle begins:
  1;
  1, 3;
  1, 5, 11;
  1, 13, 31, 55;
  1, 49, 121, 217, 337;
  1, 241, 601, 1081, 1681, 2401;
  ...

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, T[n, k-1] +(k+1)*n!];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
function T(n, k) // T = A105064
  if k eq 0 then return 1;
  else return T(n, k-1) + (k+1)*Factorial(n);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2023
(SageMath)
def T(n, k): # T = A105064
    if (k==0): return 1
    else: return T(n, k-1) + (k+1)*factorial(n)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 12 2023

Crossrefs

Sequence in context: A063853 A219078 A266033 * A073496 A176122 A370380

Adjacent sequences: A105061 A105062 A105063 * A105065 A105066 A105067

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 05 2005

Extensions

Edited by G. C. Greubel, Mar 12 2023

Status

approved