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

1, 4, 2, 11, 22, 3, 26, 132, 78, 4, 57, 604, 906, 228, 5, 120, 2382, 7248, 4764, 600, 6, 247, 8586, 46857, 62476, 21465, 1482, 7, 502, 29216, 264702, 624760, 441170, 87648, 3514, 8, 1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 24 2008

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
{   1};
{   4,     2};
{  11,    22,       3};
{  26,   132,      78,       4};
{  57,   604,     906,     228,       5};
{ 120,  2382,    7248,    4764,     600,       6};
{ 247,  8586,   46857,   62476,   21465,    1482,      7};
{ 502, 29216,  264702,  624760,  441170,   87648,   3514,   8};
{1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9}.
		

Crossrefs

Cf. A173018, A001286 (row sums).

Programs

Maple

T := (n, k) -> k * combinat:-eulerian1(n+1, k):
for n from 1 to 9 do seq(T(n, k), k = 1..n) od; # Peter Luschny, Feb 07 2023

Mathematica

T[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
Table[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], {n, 1, 10}];
Table[CoefficientList[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], x], {n, 1, 10}];
Flatten[%]
(* Alternative: *) Needs["Combinatorica`"]
Flatten[Table[k*Eulerian[n+1, k], {n, 1, 9}, {k, 1, n}]] (* Peter Luschny, Feb 07 2023 *)

Formula

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).

Extensions

Edited by Peter Luschny, Feb 07 2023