A201941 -id:A201941 - cp's OEIS frontend

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..396

Cf. A000290, A006153, A033453, A033462, A302189, A308862.

nmax = 19; CoefficientList[Series[1/(1 - x (1 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + x)*exp(x)))) \\ Michel Marcus, Mar 10 2022

E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - Vaclav Kotesovec, Jun 29 2019

A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 13 2011

For some choices of a, b, c, there is a unique value of x satisfying a*x^2+bx+c=e^x; for other choices, there are two solutions; and for others, three.  Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 0.... A126583
2.... 0.... 0.... A201936, A201937, A201938
1.... 0... -1.... A201940
1.... 1.... 0.... A201941
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A201936, take f(x,u,v)=u*x^2+v-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			least x:  -2.617866613066812769178978059143202...
greatest negative x:  -1.487962065498177156254...
greatest x:  0.5398352769028200492118039083633...

Index entries for transcendental numbers.

Cf. A201741 [a*x^2+b*x+c=e^x].

a = 2; b = 0; c = 0;
f[x_] := a*x^2 + b*x + c; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3, -2}, WorkingPrecision -> 110]
RealDigits[r]  (* A201936 *)
r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
RealDigits[r]   (* A201937 *)
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]   (* A201938 *)
(* Program 2: implicit surface of u*x^2+v=e^(-x) *)
f[{x_, u_, v_}] := u*x^2 + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, -4, 0}, {u, 1,10}];
ListPlot3D[Flatten[t, 1]]  (* for A201936 *)

A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.

Original entry on oeis.org

1, 5, 23, 154, 1389, 15636, 211231, 3329264, 59969097, 1215233380, 27362096211, 677690995488, 18310602210445, 535964033279780, 16894811428737495, 570603293774677696, 20556251540382371217, 786832900592755991364, 31889277719673937849243, 1364231113649221829763200

Offset: 1

Ilya Gutkovskiy, Jul 10 2020

nonn

Cf. A000290, A033462, A033464, A305990, A308861, A336184.

a[n_] := a[n] = n^2 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

E.g.f.: -log(1 - exp(x) * x * (1 + x)).
E.g.f.: -log(1 - Sum_{k>=1} k^2 * x^k / k!).
a(n) ~ (n-1)! / r^n, where r = A201941 = 0.444130228823966590585466329490984667... is the root of the equation exp(r)*r*(1+r) = 1. - Vaclav Kotesovec, Jul 11 2020

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A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

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A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).

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A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.

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cp's OEIS Frontend

A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

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Mathematica

PARI

Formula

A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.

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Author

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Mathematica

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A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).