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A218017 Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.

%I A218017 #37 Sep 08 2022 08:46:04
%S A218017 1,7,1,49,14,2,343,147,42,6,2401,1372,588,168,24,16807,12005,6860,
%T A218017 2940,840,120,117649,100842,72030,41160,17640,5040,720,823543,823543,
%U A218017 705894,504210,288120,123480,35280,5040,5764801,6588344,6588344,5647152,4033680,2304960,987840,282240,40320
%N A218017 Triangle, read by rows, where T(n,k) = k!*C(n, k)*7^(n-k) for n>=0, k=0..n.
%C A218017 Triangle formed by the derivatives of x^n evaluated at x=7. Also:
%C A218017 first column:    A000420;
%C A218017 second column:   A027473;
%C A218017 third column:  2*A027474;
%C A218017 fourth column: 6*A140107.
%H A218017 Vincenzo Librandi, <a href="/A218017/b218017.txt">Rows n = 0..100, flattened</a>
%F A218017 T(n,k) = 7^(n-k)*n!/(n-k)! for n>=0, k=0..n.
%F A218017 E.g.f. (by columns): exp(7x)*x^k.
%e A218017 Triangle begins:
%e A218017 1;
%e A218017 7,       1;
%e A218017 49,      14,      2;
%e A218017 343,     147,     42,      6;
%e A218017 2401,    1372,    588,     168,     24;
%e A218017 16807,   12005,   6860,    2940,    840,     120;
%e A218017 117649,  100842,  72030,   41160,   17640,   5040,    720;
%e A218017 823543,  823543,  705894,  504210,  288120,  123480,  35280,  5040; etc.
%t A218017 Flatten[Table[n!/(n-k)!*7^(n-k), {n, 0, 10}, {k, 0, n}]]
%o A218017 (Magma) [Factorial(n)/Factorial(n-k)*7^(n-k): k in [0..n], n in [0..10]];
%Y A218017 Cf. A000420, A027466, A027473, A027474, A090802, A140107, A217629, A218016.
%K A218017 nonn,tabl,easy
%O A218017 0,2
%A A218017 _Vincenzo Librandi_, Nov 10 2012