A395642 - OEIS

%I #9 Jun 16 2026 00:38:53

%S 1,1,-2,2,-8,4,6,-40,36,-8,24,-256,320,-128,16,120,-2048,3200,-1760,

%T 400,-32,720,-19968,36736,-24960,7840,-1152,64,5040,-231936,482944,

%U -382592,148960,-30464,3136,-128,40320,-3145728,7208960,-6422528,2924544,-745472

%N 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.

%e First eight rows:

%e      1

%e      1       -2

%e      2       -8        4

%e      6      -40       36       -8

%e     24     -256      320     -128       16

%e    120    -2048     3200    -1760      400     -32

%e    720   -19968    36736   -24960     7840   -1152     64

%e   5040  -231936   482944  -382592   148960  -30464   3136  -128

%e Row 4 represents the polynomial 24 - 256*x + 320*x^2 - 128*x^3 + 16*x^4.

%t s[n_] := Join[{{0, -1}}, Table[{k, (-1)^(k + 1)}, {k, 1, n}]];

%t p[n_, x_] := Expand[InterpolatingPolynomial[s[n], x]];

%t Column[Table[CoefficientList[-n!*p[n, x], x], {n, 0, 10}]]

%t Flatten[Table[CoefficientList[-n!*p[n, x], x], {n, 0, 10}]]

%t 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."]

%Y Cf. A395640, A395641.

%K tabl,sign

%O 0,3

%A _Clark Kimberling_, Jun 03 2026