A395640 Triangular array: T(n, k) is the coefficient of x^k in the polynomial - n!*L(n, x), where L(n, x) is the Lagrange interpolating polynomial that passes through (1,0) and (k,1) for k = 2..n+1.

0, 1, -1, 4, -5, 1, 18, -26, 9, -1, 96, -154, 71, -14, 1, 600, -1044, 580, -155, 20, -1, 4320, -8028, 5104, -1665, 295, -27, 1, 35280, -69264, 48860, -18424, 4025, -511, 35, -1, 322560, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, 3265920, -6999840

Offset

0,4

Links

Table of n, a(n) for n=0..46.

Example

First eight rows:
      0
      1      -1
      4      -5      1
     18     -26      9      -1
     96    -154     71     -14     1
    600   -1044    580    -155    20    -1
   4320   -8028   5104   -1665   295   -27   1
  35280  -69264  48860  -18424  4025  -511  35   -1
L(4, x) = -4 + (77 x)/12 - (71 x^2)/24 + (7 x^3)/12 - x^4/24,
so 4!*L (4, x) = -96 + 154 x - 71 x^2 + 14 x^3 - x^4, whence row 4 of the array is (96, -154, 71, -14, 1).

Mathematica

t[0] = {0}; t[n_] := t[n] = Join[t[n - 1], {1}];
p[n_, x_] := Expand[InterpolatingPolynomial[t[n], x]];
Column[Join[{{0}}, Table[CoefficientList[-n!*p[n, x], x], {n, 1, 10}]]]
Join[{0}, Flatten[Table[CoefficientList[-n!*p[n, x], x], {n, 0, 10}]]]
Labeled[Plot[{p[1, x], p[2, x], p[3, x], p[4, x], p[5, x]}, {x, -1, 7}], "Plot of p[n, x] for n=1..5."]

Crossrefs

Cf. A001563 (column 1), A001701 (column 2), A001706 (column 3), A001797 (column 4), A395641, A395642, A004442, A125577, A000531, A000346, A396722, A396723, A396724.

Sequence in context: A196848 A369950 A266699 * A234937 A210590 A108446

Adjacent sequences: A395637 A395638 A395639 * A395641 A395642 A395643

Keyword

tabl,sign,new

Author

Clark Kimberling, Jun 03 2026

Extensions

Name edited by Peter Munn, Jun 16 2026

Status

approved