A395641 Triangular array: row n shows the coefficients of the polynomial - n!*L(n, x), where L(n, x) is the Lagrange interpolating polynomial that passes through (0,-1) and (k,1) for k = 1..n.

1, 1, -2, 2, -6, 2, 6, -22, 12, -2, 24, -100, 70, -20, 2, 120, -548, 450, -170, 30, -2, 720, -3528, 3248, -1470, 350, -42, 2, 5040, -26136, 26264, -13538, 3920, -644, 56, -2, 40320, -219168, 236248, -134568, 44898, -9072, 1092, -72, 2, 362880, -2053152

Offset

0,3

Links

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

Example

First eight rows:
      1
      1    -2
      2    -6       2
      6    -22     12     -2
     24   -100     70    -20     2
    120   -548    450   -170    30   -2
    720  -3528   3248  -1470   350  -42   2
   5040 -26136  26264 -13538  3920 -644  56  -2
Row 4 represents the polynomial 24 - 100*x + 70*x^2 - 20*x^3 + 2*x^4.

Mathematica

s[n_] := Join[{{0, -1}}, Table[{k, 1}, {k, 1, n}]];
p[n_, x_] := Expand[InterpolatingPolynomial[s[n], x]];
Column[Table[CoefficientList[-n!*p[n, x], x], {n, 0, 10}]]
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. A000142 (column 1), A395640, A395642.

Sequence in context: A257257 A257251 A068555 * A167556 A221438 A327991

Adjacent sequences: A395638 A395639 A395640 * A395642 A395643 A395644

Keyword

tabl,sign

Author

Clark Kimberling, Jun 03 2026

Status

approved