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A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

%I A217629 #41 Sep 08 2022 08:46:04
%S A217629 1,3,1,9,6,2,27,27,18,6,81,108,108,72,24,243,405,540,540,360,120,729,
%T A217629 1458,2430,3240,3240,2160,720,2187,5103,10206,17010,22680,22680,15120,
%U A217629 5040,6561,17496,40824,81648,136080,181440,181440,120960,40320
%N A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.
%C A217629 Triangle formed by the derivatives of x^n evaluated at x=3.
%C A217629 Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:
%C A217629 first column:     A000244;
%C A217629 second column:    A027471;
%C A217629 third column:   2*A027472;
%C A217629 fourth column:  6*A036216;
%C A217629 fifth column:  24*A036217.
%H A217629 Vincenzo Librandi, <a href="/A217629/b217629.txt">Rows n = 0..100, flattened</a>
%F A217629 T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.
%F A217629 E.g.f. (by columns): exp(3x)*x^k.
%e A217629 Triangle begins:
%e A217629 1;
%e A217629 3,     1;
%e A217629 9,     6,     2;
%e A217629 27,    27,    18,     6;
%e A217629 81,    108,   108,    72,     24;
%e A217629 243,   405,   540,    540,    360,    120;
%e A217629 729,   1458,  2430,   3240,   3240,   2160,    720;
%e A217629 2187,  5103,  10206,  17010,  22680,  22680,   15120,   5040;
%e A217629 6561,  17496, 40824,  81648,  136080, 181440,  181440,  120960,  40320; etc.
%t A217629 Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]
%o A217629 (Magma) [Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];
%Y A217629 Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.
%K A217629 nonn,tabl,easy
%O A217629 0,2
%A A217629 _Vincenzo Librandi_, Nov 10 2012