A174689 Triangle T(n, k) = n! * binomial(n, k)^2 - n! + 1, read by rows.

1, 1, 1, 1, 7, 1, 1, 49, 49, 1, 1, 361, 841, 361, 1, 1, 2881, 11881, 11881, 2881, 1, 1, 25201, 161281, 287281, 161281, 25201, 1, 1, 241921, 2217601, 6168961, 6168961, 2217601, 241921, 1, 1, 2540161, 31570561, 126403201, 197527681, 126403201, 31570561, 2540161, 1

Offset

0,5

Links

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

Formula

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

From G. C. Greubel, Feb 10 2021: (Start)

T(n, k) = n! * ( A008459(n, k) - 1 ) + 1.

Sum_{k=0..n} T(n, k) = (n+1)*( n!*( C_{n} - 1 ) + 1 ) = (n+1)*( n!*( A000108(n) - 1 ) + 1). (End)

Example

Triangle begins as:
  1;
  1,       1;
  1,       7,        1;
  1,      49,       49,         1;
  1,     361,      841,       361,         1;
  1,    2881,    11881,     11881,      2881,         1;
  1,   25201,   161281,    287281,    161281,     25201,        1;
  1,  241921,  2217601,   6168961,   6168961,   2217601,   241921,       1;
  1, 2540161, 31570561, 126403201, 197527681, 126403201, 31570561, 2540161, 1;

Mathematica

T[n_, k_]:= n!*Binomial[n, k]^2 - n! + 1;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage) flatten([[factorial(n)*(binomial(n, k)^2 -1) + 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021
(Magma) [Factorial(n)*(Binomial(n, k)^2 -1) + 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Crossrefs

Cf. A000108, A008459, A174690.

Sequence in context: A178658 A156602 A203389 * A172345 A174719 A176392

Adjacent sequences: A174686 A174687 A174688 * A174690 A174691 A174692

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 27 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved