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%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
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