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A142706 Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).

%I A142706 #10 Feb 07 2023 12:43:13
%S A142706 1,4,2,11,22,3,26,132,78,4,57,604,906,228,5,120,2382,7248,4764,600,6,
%T A142706 247,8586,46857,62476,21465,1482,7,502,29216,264702,624760,441170,
%U A142706 87648,3514,8,1013,95680,1365576,5241416,6551770,2731152,334880,8104,9
%N A142706 Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).
%F A142706 Let E(n, x) = Sum_{j=0..k} A173018(n, k)*x^k and E'(n, x) = (d/dx) E(x, n). Then T(n, k) = [x^(k-1)] E'(n+1, x).
%e A142706 Triangle T(n, k) starts:
%e A142706 {   1};
%e A142706 {   4,     2};
%e A142706 {  11,    22,       3};
%e A142706 {  26,   132,      78,       4};
%e A142706 {  57,   604,     906,     228,       5};
%e A142706 { 120,  2382,    7248,    4764,     600,       6};
%e A142706 { 247,  8586,   46857,   62476,   21465,    1482,      7};
%e A142706 { 502, 29216,  264702,  624760,  441170,   87648,   3514,   8};
%e A142706 {1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9}.
%p A142706 T := (n, k) -> k * combinat:-eulerian1(n+1, k):
%p A142706 for n from 1 to 9 do seq(T(n, k), k = 1..n) od; # _Peter Luschny_, Feb 07 2023
%t A142706 T[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%t A142706 Table[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], {n, 1, 10}];
%t A142706 Table[CoefficientList[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], x], {n, 1, 10}];
%t A142706 Flatten[%]
%t A142706 (* Alternative: *) Needs["Combinatorica`"]
%t A142706 Flatten[Table[k*Eulerian[n+1, k], {n, 1, 9}, {k, 1, n}]] (* _Peter Luschny_, Feb 07 2023 *)
%Y A142706 Cf. A173018, A001286 (row sums).
%K A142706 nonn,tabl
%O A142706 1,2
%A A142706 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2008
%E A142706 Edited by _Peter Luschny_, Feb 07 2023