cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A213640 E.g.f. satisfies: A(x) = x - log(1 - A(x)^2).

Original entry on oeis.org

1, 2, 12, 132, 2040, 40560, 986160, 28344960, 940222080, 35350378560, 1485586206720, 69006955691520, 3510875174526720, 194162144086310400, 11597083480958976000, 744005639375065267200, 51024015181398702643200, 3725042532308649628876800, 288434288836744276094668800
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2012

Keywords

Examples

			E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
Related series:
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1800*x^5/5! + 35280*x^6/6! +...
-log(1-A(x)^2) = 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...

Crossrefs

Cf. A213641, A218652, A200319.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x + Log[1-x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x+log(1-x^2 +x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (-log(1 - x^2+x*O(x^n)))^m)/m!); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (-log(1 - x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x + log(1-x^2)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n/x / n! ).
a(n) = n*A213641(n-1).
a(n) ~ sqrt(1-1/sqrt(2)) * n^(n-1) / (exp(n) * (sqrt(2) -1 + log(2*(sqrt(2)-1)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014

A218653 E.g.f. satisfies: A(x) = 1 + log(1 + x^2*A(x)^2)/x.

Original entry on oeis.org

1, 1, 4, 27, 264, 3400, 54480, 1045800, 23412480, 599157216, 17258814720, 552733695360, 19485393903360, 749871707270400, 31283408387911680, 1406370859616923200, 67780975948945459200, 3486485719168394342400, 190644828634476331315200, 11043310871932837194977280
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2012

Keywords

Examples

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 264*x^4/4! + 3400*x^5/5! +...
Related expansions:
A(x)^2 = 1 + 2*x + 10*x^2/2! + 78*x^3/3! + 840*x^4/4! + 11600*x^5/5! +...
log(1 + x^2*A(x)^2)/x = x + 4*x^2/2! + 27*x^3/3! + 264*x^4/4! + 3400*x^5/5! +...

Links

Crossrefs

Cf. A218652, A213641.

Programs

  • PARI
    {a(n)=n!*polcoeff((1/x)*serreverse(x-log(1+x^2 +x^2*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. satisfies: A(x - log(1+x^2)) = x/(x - log(1+x^2)).
E.g.f.: A(x) = (1/x)*Series_Reversion(x - log(1+x^2)).
a(n) = A218652(n+1)/(n+1).
a(n) ~ Gamma(1/3) * n^(n - 5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (1 - log(2))^(n + 2/3)). - Vaclav Kotesovec, Oct 07 2020

A228173 E.g.f. satisfies: A(x) = 1+x - exp(-A(x)^2).

Original entry on oeis.org

1, 2, 12, 108, 1320, 20280, 374640, 8072400, 198465120, 5475284640, 167285321280, 5600184004800, 203602252613760, 7978382871338880, 334767145102790400, 14952953514231532800, 707221717016278464000, 35242469168705967168000, 1841491290250262851200000
Offset: 1

Views

Author

Paul D. Hanna, Aug 14 2013

Keywords

Comments

Note that a(30) is negative. - Vaclav Kotesovec, Sep 16 2013

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...
where
exp(-A(x)^2) = 1 - 2*x^2/2! - 12*x^3/3! - 108*x^4/4! - 1320*x^5/5! -...
The e.g.f. equals the series:
A(x) = x + (1 - exp(-x^2)) + d/dx (1 - exp(-x^2))^2/2! + d^2/dx^2 (1 - exp(-x^2))^3/3! + d^3/dx^3 (1 - exp(-x^2))^4/4! + d^4/dx^4 (1 - exp(-x^2))^5/5! +...
Also,
log(A(x)/x) = (1 - exp(-x^2))/x + d/dx (1 - exp(-x^2))^2/(2!*x) + d^2/dx^2 (1 - exp(-x^2))^3/(3!*x) + d^3/dx^3 (1 - exp(-x^2))^4/(4!*x) +...

Links

Crossrefs

Cf. A218652.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x-1+E^(-x^2),{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Sep 16 2013 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-1+exp(-x^2+x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (1 - exp(-x^2+x*O(x^n)))^m)/m!); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (1 - exp(-x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x - 1 + exp(-x^2)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1 - exp(-x^2))^n/x / n! ).
Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1/abs(-1-(LambertW(-1/2)-1) / sqrt(-2*LambertW(-1/2))) = 3.19002880735268... - Vaclav Kotesovec, Jan 11 2014
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.