A257395 Decimal expansion of the y-coordinate of the inflection point of product{1 + x^k, k >= 1} that has minimal x-coordinate.

1, 7, 6, 7, 1, 9, 5, 7, 8, 5, 6, 2, 9, 3, 7, 4, 3, 4, 7, 5, 7, 2, 1, 5, 9, 2, 6, 1, 8, 5, 7, 4, 1, 8, 6, 1, 1, 2, 5, 3, 7, 8, 0, 2, 9, 9, 3, 1, 3, 2, 0, 1, 3, 4, 9, 7, 3, 2, 3, 5, 3, 7, 2, 4, 8, 1, 3, 2, 5, 5, 7, 2, 6, 6, 2, 8, 1, 8, 4, 1, 4, 2, 8, 5, 7, 7, 3, 1, 8, 2, 3, 5, 1, 4, 7, 8, 6, 5, 7, 8, 1, 1, 0, 4

Offset

0,2

Comments

The function product{1 + x^k, k >= 1} has two inflection points: (-0.78983..., 0.17671...) and (-0.23233..., 0.80084...).

Links

Table of n, a(n) for n=0..103.

Example

y = 0.1767195785629374347572159261857418...

Mathematica

f[x_] := f[x] = Product[(1 + x^k), {k, 1, 1000}];
p[x_, z_] := Sum[n/(x + x^(1 - n)), {n, z}]^2 + Sum[(n*x^(n - 2)*(n - x^n - 1))/(1 + x^n)^2, {n, z}];
Plot[f[x], {x, -1, 1}] (* plot showing 2 infl. pts. *)
t = x /. FindRoot[p[x, 1000], {x, -0.8}, WorkingPrecision -> 100] (* A257394 *)
u = f[t] (* A257395 *)
v = x /. FindRoot[p[x, 200], {x, -0.3}, WorkingPrecision -> 100]  (* A257396 *)
w = f[v] (* A257397 *)
RealDigits[t, 10][[1]]  (* A257394 *)
RealDigits[u, 10][[1]]  (* A257395 *)
RealDigits[v, 10][[1]]  (* A257396 *)
RealDigits[w, 10][[1]]  (* A257397 *)
(* Peter J. C. Moses, Apr 21 2015 *)
digits = 104; QP = QPochhammer; QPP[x_] := With[{dx = 10^-digits}, (QP[-1, x+dx] - QP[-1, x-dx ])/(4*dx)]; x0 = x /. NMaximize[{QPP[x], -1 < x < -1/2}, x, WorkingPrecision -> 4 digits][[2]]; y = QP[-1, x0]/2; RealDigits[y, 10, digits][[1]] (* Jean-François Alcover, Nov 19 2015 *)

Crossrefs

Cf. A257394, A257396, A257397.

Sequence in context: A021570 A242977 A247314 * A197687 A239341 A239606

Adjacent sequences: A257392 A257393 A257394 * A257396 A257397 A257398

Keyword

nonn,cons,easy

Author

Clark Kimberling, Apr 22 2015

Extensions

More digits from Jean-François Alcover, Nov 19 2015

Status

approved