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A257397 Decimal expansion of the y-coordinate of the inflection point of product{1 + x^k, k >= 1} that has maximal x-coordinate. 4
8, 0, 0, 8, 4, 1, 5, 5, 3, 3, 8, 8, 0, 4, 5, 4, 5, 8, 4, 6, 4, 2, 2, 8, 3, 3, 4, 2, 5, 6, 8, 3, 5, 1, 3, 4, 2, 0, 2, 5, 9, 7, 7, 7, 6, 6, 0, 0, 0, 5, 3, 5, 5, 3, 0, 6, 5, 3, 1, 1, 7, 0, 3, 4, 5, 6, 2, 9, 1, 5, 3, 7, 2, 6, 3, 7, 9, 9, 7, 7, 9, 8, 7, 3, 4, 2, 8, 4, 2, 3, 5, 8, 9, 2, 1, 8, 5, 8, 6, 7, 7, 7, 5, 7, 7, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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..106.

EXAMPLE

y = 0.80084155338804545846422833425683513420259777660005...

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 = 107; QP = QPochhammer; QPP[x_] := With[{dx = 10^-digits}, (QP[-1, x+dx] - QP[-1, x-dx])/(4*dx)]; x0 = x /. NMinimize[{QPP[x], -1/2 < x < 0}, x, WorkingPrecision -> 4 digits][[2]]; y = QP[-1, x0]/2; RealDigits[y, 10, digits][[1]] (* Jean-François Alcover, Nov 19 2015 *)

CROSSREFS

Cf. A257394, A257395, A257396.

Sequence in context: A167261 A114611 A219241 * A075448 A271407 A347801

Adjacent sequences:  A257394 A257395 A257396 * A257398 A257399 A257400

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Apr 22 2015

EXTENSIONS

More digits from Jean-François Alcover, Nov 19 2015

STATUS

approved

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Last modified October 23 23:05 EDT 2021. Contains 348217 sequences. (Running on oeis4.)