A167040 - OEIS

%I #7 Feb 25 2021 02:38:37

%S 1,1,1,1,2,1,1,3,3,1,1,4,96,4,1,1,5,320,320,5,1,1,6,960,14580,960,6,1,

%T 1,7,2688,76545,76545,2688,7,1,1,8,7168,367416,4587520,367416,7168,8,

%U 1,1,9,18432,1653372,33030144,33030144,1653372,18432,9,1

%N Triangle T(n, k) = (n-k)^n * binomial(n, n-k) for n < 2*k, k^n * binomial(n, k) for n >= 2*k with T(n, 0) = T(n, n) = 1, read by rows.

%C Row sums are: 1, 2, 4, 8, 106, 652, 16514, 158482, 5336706, 69403916, 2915603362, ...

%H G. C. Greubel, <a href="/A167040/b167040.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Feb 24 2021: (Start)

%F T(n, k) = (n-k)^n * binomial(n, n-k) for n < 2*k, k^n * binomial(n, k) for n >= 2*k with T(n, 0) = T(n, n) = 1.

%F T(n, k) = T(n, n-k). (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  2,     1;

%e   1,  3,     3,       1;

%e   1,  4,    96,       4,         1;

%e   1,  5,   320,     320,         5,          1;

%e   1,  6,   960,   14580,       960,          6,         1;

%e   1,  7,  2688,   76545,     76545,       2688,         7,       1;

%e   1,  8,  7168,  367416,   4587520,     367416,      7168,       8,     1;

%e   1,  9, 18432, 1653372,  33030144,   33030144,   1653372,   18432,     9,  1;

%e   1, 10, 46080, 7085880, 220200960, 2460937500, 220200960, 7085880, 46080, 10, 1;

%t T[n_, k_]:= If[k==0 || k==n, 1, If[n>=2*k, k^n*Binomial[n, k], (n-k)^n*Binomial[n, n-k]]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 24 2021 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k==0 or k==n): return 1

%o     elif (n > 2*k-1): return k^n*binomial(n,k)

%o     else: return (n-k)^n*binomial(n,n-k)

%o flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 24 2021

%o (Magma)

%o function T(n,k)

%o   if k eq 0 or k eq n then return 1;

%o   elif n lt 2*k then return (n-k)^n*Binomial(n,n-k);

%o   else return k^n*Binomial(n,k);

%o   end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 24 2021

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_ and _Mats Granvik_, Oct 27 2009

%E Edited by _G. C. Greubel_, Feb 24 2021