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

1,2

Links

Table of n, a(n) for n=1..45.

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).

Example

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}.

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 *)

Crossrefs

Cf. A173018, A001286 (row sums).

Sequence in context: A185878 A182870 A094406 * A361216 A092952 A286145

Adjacent sequences: A142703 A142704 A142705 * A142707 A142708 A142709

Keyword

nonn,tabl

Author

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

Extensions

Edited by Peter Luschny, Feb 07 2023

Status

approved