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A218016 Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

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