A137259 - OEIS

%I #22 Jan 05 2024 12:56:11

%S 0,0,0,3,3,0,20,20,16,0,115,115,110,90,0,714,714,708,684,576,0,5033,

%T 5033,5026,4998,4872,4200,0,40312,40312,40304,40272,40128,39360,34560,

%U 0,362871,362871,362862,362826,362664,361800,356400,317520,0,3628790,3628790,3628780,3628740,3628560,3627600,3621600,3578400,3225600,0

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

%H G. C. Greubel, <a href="/A137259/b137259.txt">Rows n = 1..50 of the triangle, flattened</a>

%H Krassimir Penev, <a href="https://olympiadtraining.files.wordpress.com/2014/10/penev-the-fubini-principle.pdf">The Fubini Principle</a>, The American Mathematical Monthly, Vol. 115, No. 3 (Mar., 2008), pp. 245-248.

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

%F Sum_{k=1..n} T(n, k) = n*(n! - !n) = n*(n! - A003422(n)). - _G. C. Greubel_, Apr 10 2021

%e Triangle begins as:

%e        0;

%e        0,      0;

%e        3,      3,      0;

%e       20,     20,     16,      0;

%e      115,    115,    110,     90,      0;

%e      714,    714,    708,    684,    576,      0;

%e     5033,   5033,   5026,   4998,   4872,   4200,      0;

%e    40312,  40312,  40304,  40272,  40128,  39360,  34560,      0;

%e   362871, 362871, 362862, 362826, 362664, 361800, 356400, 317520, 0;

%p 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

%t T[n_, k_]:= n! - n*(k-1)!;

%t Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Apr 10 2021 *)

%o (Magma) [n*(Factorial(n-1) - Factorial(k-1)): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 10 2021

%o (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

%Y Cf. A003422, A137260.

%K nonn,tabl,easy

%O 1,4

%A _Roger L. Bagula_, Mar 11 2008

%E Edited by _G. C. Greubel_, Apr 10 2021