A137259 Triangle T(n, k) = n*( (n-1)! - (k-1)! ), read by rows.

0, 0, 0, 3, 3, 0, 20, 20, 16, 0, 115, 115, 110, 90, 0, 714, 714, 708, 684, 576, 0, 5033, 5033, 5026, 4998, 4872, 4200, 0, 40312, 40312, 40304, 40272, 40128, 39360, 34560, 0, 362871, 362871, 362862, 362826, 362664, 361800, 356400, 317520, 0, 3628790, 3628790, 3628780, 3628740, 3628560, 3627600, 3621600, 3578400, 3225600, 0

Offset

1,4

Links

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

Krassimir Penev, The Fubini Principle, The American Mathematical Monthly, Vol. 115, No. 3 (Mar., 2008), pp. 245-248.

Formula

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

Sum_{k=1..n} T(n, k) = n*(n! - !n) = n*(n! - A003422(n)). - G. C. Greubel, Apr 10 2021

Example

Triangle begins as:
       0;
       0,      0;
       3,      3,      0;
      20,     20,     16,      0;
     115,    115,    110,     90,      0;
     714,    714,    708,    684,    576,      0;
    5033,   5033,   5026,   4998,   4872,   4200,      0;
   40312,  40312,  40304,  40272,  40128,  39360,  34560,      0;
  362871, 362871, 362862, 362826, 362664, 361800, 356400, 317520, 0;

Maple

A137259:= (n, k) -> n*((n-1)! - (k-1)!); seq(seq(A137259(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 10 2021

Mathematica

T[n_, k_]:= n! - n*(k-1)!;
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 10 2021 *)

Prog

(Magma) [n*(Factorial(n-1) - Factorial(k-1)): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 10 2021
(Sage) flatten([[n*(factorial(n-1) - factorial(k-1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 10 2021

Crossrefs

Cf. A003422, A137260.

Sequence in context: A374828 A247889 A309012 * A166553 A285863 A111843

Adjacent sequences: A137256 A137257 A137258 * A137260 A137261 A137262

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 11 2008

Extensions

Edited by G. C. Greubel, Apr 10 2021

Status

approved