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Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.
4

%I #41 Sep 08 2022 08:46:04

%S 1,3,1,9,6,2,27,27,18,6,81,108,108,72,24,243,405,540,540,360,120,729,

%T 1458,2430,3240,3240,2160,720,2187,5103,10206,17010,22680,22680,15120,

%U 5040,6561,17496,40824,81648,136080,181440,181440,120960,40320

%N Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

%C Triangle formed by the derivatives of x^n evaluated at x=3.

%C Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:

%C first column: A000244;

%C second column: A027471;

%C third column: 2*A027472;

%C fourth column: 6*A036216;

%C fifth column: 24*A036217.

%H Vincenzo Librandi, <a href="/A217629/b217629.txt">Rows n = 0..100, flattened</a>

%F T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.

%F E.g.f. (by columns): exp(3x)*x^k.

%e Triangle begins:

%e 1;

%e 3, 1;

%e 9, 6, 2;

%e 27, 27, 18, 6;

%e 81, 108, 108, 72, 24;

%e 243, 405, 540, 540, 360, 120;

%e 729, 1458, 2430, 3240, 3240, 2160, 720;

%e 2187, 5103, 10206, 17010, 22680, 22680, 15120, 5040;

%e 6561, 17496, 40824, 81648, 136080, 181440, 181440, 120960, 40320; etc.

%t Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]

%o (Magma) [Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];

%Y Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.

%K nonn,tabl,easy

%O 0,2

%A _Vincenzo Librandi_, Nov 10 2012