%I #45 Sep 08 2022 08:46:04
%S 1,5,1,25,10,2,125,75,30,6,625,500,300,120,24,3125,3125,2500,1500,600,
%T 120,15625,18750,18750,15000,9000,3600,720,78125,109375,131250,131250,
%U 105000,63000,25200,5040,390625,625000,875000,1050000,1050000,840000,504000,201600,40320
%N Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.
%C Triangle formed by the derivatives of x^n evaluated at x=5.
%C Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:
%C first column: A000351;
%C second column: A053464;
%C third column: 2*A084902;
%C fourth column: 6*A081143.
%H Vincenzo Librandi, <a href="/A218016/b218016.txt">Rows n = 0..100, flattened</a>
%F T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.
%F E.g.f. (by columns): exp(5x)*x^k.
%e Triangle begins:
%e 1;
%e 5, 1;
%e 25, 10, 2;
%e 125, 75, 30, 6;
%e 625, 500, 300, 120, 24;
%e 3125, 3125, 2500, 1500, 600, 120;
%e 15625, 18750, 18750, 15000, 9000, 3600, 720;
%e 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040;
%e 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.
%t Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]
%o (Magma) [Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];
%Y Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.
%K nonn,tabl,easy
%O 0,2
%A _Vincenzo Librandi_, Nov 10 2012