A173476 Triangle T(n, k) = 1 + (k!)^2 - 2*k!*(n-k)! + ((n-k)!)^2, read by rows.

1, 1, 1, 2, 1, 2, 26, 2, 2, 26, 530, 26, 1, 26, 530, 14162, 530, 17, 17, 530, 14162, 516962, 14162, 485, 1, 485, 14162, 516962, 25391522, 516962, 13925, 325, 325, 13925, 516962, 25391522, 1625621762, 25391522, 515525, 12997, 1, 12997, 515525, 25391522, 1625621762

Offset: 0

Author

Roger L. Bagula, Feb 19 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
           1;
           1,        1;
           2,        1,      2;
          26,        2,      2,    26;
         530,       26,      1,    26, 530;
       14162,      530,     17,    17, 530, 14162;
      516962,    14162,    485,     1, 485, 14162, 516962;
    25391522,   516962,  13925,   325, 325, 13925, 516962, 25391522;
  1625621762, 25391522, 515525, 12997,   1, 12997, 515525, 25391522, 1625621762;
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A003149, A061062.

Programs

Magma

[(Factorial(n-k) -Factorial(k))^2 +1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021

Mathematica

Table[((n-k)! -k!)^2 +1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)

Sage

flatten([[(factorial(n-k) -factorial(k))^2 +1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021

Formula

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

Sum_{k=0..n} T(n, k) = 1 + n + 2*A061062(n) - 2*A003149(n). - G. C. Greubel, Feb 19 2021

Extensions

Edited by G. C. Greubel, Feb 19 2021