A395642 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)^(k+1)) for k = 1..n.

1, 1, -2, 2, -8, 4, 6, -40, 36, -8, 24, -256, 320, -128, 16, 120, -2048, 3200, -1760, 400, -32, 720, -19968, 36736, -24960, 7840, -1152, 64, 5040, -231936, 482944, -382592, 148960, -30464, 3136, -128, 40320, -3145728, 7208960, -6422528, 2924544, -745472

Offset

0,3

Links

Table of n, a(n) for n=0..41.

Example

First eight rows:
     1
     1       -2
     2       -8        4
     6      -40       36       -8
    24     -256      320     -128       16
   120    -2048     3200    -1760      400     -32
   720   -19968    36736   -24960     7840   -1152     64
  5040  -231936   482944  -382592   148960  -30464   3136  -128
Row 4 represents the polynomial 24 - 256*x + 320*x^2 - 128*x^3 + 16*x^4.

Mathematica

s[n_] := Join[{{0, -1}}, Table[{k, (-1)^(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. A395640, A395641.

Sequence in context: A369771 A239677 A331333 * A120399 A385394 A144816

Adjacent sequences: A395639 A395640 A395641 * A395643 A395644 A395645

Keyword

tabl,sign

Author

Clark Kimberling, Jun 03 2026

Status

approved