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.

A218652 E.g.f. satisfies: A(x) = x + log(1 + A(x)^2).

Original entry on oeis.org

1, 2, 12, 108, 1320, 20400, 381360, 8366400, 210712320, 5991572160, 189846961920, 6632804344320, 253310120743680, 10498203901785600, 469251125818675200, 22501933753870771200, 1152276591132072806400, 62756742945031098163200, 3622251744055050294988800
Offset: 1

Views

Author

Paul D. Hanna, Nov 03 2012

Keywords

Examples

			E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...
Related series:
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1560*x^5/5! + 25200*x^6/6! +...
log(1+A(x)^2) = 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...

Links

Crossrefs

Cf. A218653, A213640.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Log[1 + x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 19 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*A218653(n-1).
a(n) ~ GAMMA(1/3) * n^(n-5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (1-log(2))^(n-1/3)). - Vaclav Kotesovec, Jan 19 2014

A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).

Original entry on oeis.org

1, 2, 12, 132, 2040, 40440, 979440, 28034160, 925858080, 34654465440, 1449705660480, 67029745527360, 3394417068282240, 186842736763562880, 11107390768144070400, 709223357051739129600, 48408150749346010022400, 3517279496138031162739200, 271050342684747077612160000
Offset: 1

Views

Author

Paul D. Hanna, Nov 15 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
where A(1+x - exp(x^2)) = x and A(x) = x-1 + exp(A(x)^2).

Crossrefs

Cf. A200320, A213640, A202356.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[1 - E^x^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+x^2*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, (exp(x^2+x*O(x^n))-1)^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, (exp(x^2+x*O(x^n))-1)^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

Formula

E.g.f.: Series_Reversion(1+x - exp(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n/x / n! ).
a(n) ~ (c/2)^(1/4) * n^(n-1) / (sqrt(1+c) * exp(n) * (1+sqrt(c/2)-1/sqrt(2*c))^(n-1/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A213641 E.g.f. satisfies: A(x) = 1 - log(1 - x^2*A(x)^2) / x.

Original entry on oeis.org

1, 1, 4, 33, 408, 6760, 140880, 3543120, 104469120, 3535037856, 135053291520, 5750579640960, 270067321117440, 13868724577593600, 773138898730598400, 46500352460941579200, 3001412657729335449600, 206946807350480534937600, 15180752044039172426035200
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6760*x^5/5! +...
Related expansions:
A(x)^2 = 1 + 2*x + 10*x^2/2! + 90*x^3/3! + 1176*x^4/4! + 20240*x^5/5! +...
-log(1 - x^2*A(x)^2)/x = x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! +...
A(x + log(1-x^2)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 340*x^5/5! +...

Crossrefs

Cf. A213640, A218653, A200320.

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) = A213640(n+1)/(n+1).
Showing 1-3 of 3 results.

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