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

A200320 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2/2).

Original entry on oeis.org

1, 1, 3, 18, 150, 1590, 20580, 314790, 5554710, 111071520, 2482076520, 61301435580, 1658129152680, 48749053413060, 1547849157554700, 52785934927525800, 1924269399236784600, 74672595203551745400, 3073314600152521124400, 133716009695044269893400, 6132253708189762323370200
Offset: 1

Views

Author

Paul D. Hanna, Nov 15 2011

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 18*x^4/4! + 150*x^5/5! +...
where A(1+x - exp(x^2/2)) = x and A(x) = x-1 + exp(A(x)^2/2).

Crossrefs

Cf. A200319, A213641, A030178.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[1 - E^(x^2/2) + x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(1+x-exp(x^2/2+x^2*O(x^n))),n)}

Formula

E.g.f.: Series_Reversion(1+x - exp(x^2/2)).
a(n) ~ n^(n-1) * c^(n/2) / (sqrt(1+c) * exp(n) * (c-1+sqrt(c))^(n-1/2)), where c = LambertW(1) = 0.5671432904... (see A030178). - Vaclav Kotesovec, Jan 10 2014
Showing 1-3 of 3 results.

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