A137260 Triangle T(n, k) = k*(n-1)! - k!, read by rows.

0, 0, 0, 1, 2, 0, 5, 10, 12, 0, 23, 46, 66, 72, 0, 119, 238, 354, 456, 480, 0, 719, 1438, 2154, 2856, 3480, 3600, 0, 5039, 10078, 15114, 20136, 25080, 29520, 30240, 0, 40319, 80638, 120954, 161256, 201480, 241200, 277200, 282240, 0, 362879, 725758, 1088634, 1451496, 1814280, 2176560, 2535120, 2862720, 2903040, 0

Offset

1,5

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

Sum_{k=1..n} T(n, k) = ((n+1)! - 2*!(n+1))/2 = (A000142(n+1) - 2*(A003422(n+1) -1))/2 = (A000142(n+1) - 2*(A007489(n) - 2))/2. - G. C. Greubel, Apr 10 2021

Example

Triangle begins as:
       0;
       0,      0;
       1,      2,       0;
       5,     10,      12,       0;
      23,     46,      66,      72,       0;
     119,    238,     354,     456,     480,       0;
     719,   1438,    2154,    2856,    3480,    3600,       0;
    5039,  10078,   15114,   20136,   25080,   29520,   30240,       0;
   40319,  80638,  120954,  161256,  201480,  241200,  277200,  282240,       0;
  362879, 725758, 1088634, 1451496, 1814280, 2176560, 2535120, 2862720, 2903040, 0;

Maple

A137260:= (n, k) -> k*((n-1)! - (k-1)!); seq(seq(A137260(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 10 2021

Mathematica

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

Prog

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

Crossrefs

Cf. A000142, A007489.

Sequence in context: A358305 A292590 A080901 * A263101 A219765 A153059

Adjacent sequences: A137257 A137258 A137259 * A137261 A137262 A137263

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 11 2008

Extensions

Edited by G. C. Greubel, Apr 10 2021

Status

approved