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 1-4 of 4 results.

A226758 E.g.f.: A(x) = x + sin(A(x)^2).

Original entry on oeis.org

1, 2, 12, 120, 1680, 30120, 658560, 16994880, 505612800, 17037851040, 641393786880, 26678131159680, 1215016298496000, 60135628841608320, 3213908573331456000, 184463573184501811200, 11316253482729190195200, 738934748606732911833600, 51171600229826941786521600
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
where A(x - sin(x^2)) = x and A(x) = x + sin(A(x)^2).
Series expansions:
A(x) = x + sin(x^2) + d/dx sin(x^2)^2/2! + d^2/dx^2 sin(x^2)^3/3! + d^3/dx^3 sin(x^2)^4/4! +...
log(A(x)/x) = sin(x^2)/x + d/dx (sin(x^2)^2/x)/2! + d^2/dx^2 (sin(x^2)^3/x)/3! + d^3/dx^3 (sin(x^2)^4/x)/4! +...

Crossrefs

Cf. A215188, A226759, A226760.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Sin[x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 23 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-sin(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, sin(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, sin(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.: Series_Reversion(x - sin(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x^2)^n/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*sin(x^2)^n/n! ).
a(n) ~ n^(n-1) * sqrt(r/(1/s - 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.5186522338890123015... is the root of the equation 2*s*cos(s^2) = 1, and r = s - sin(s^2) = 0.2528845666082260013... - Vaclav Kotesovec, Jan 23 2014

A226759 E.g.f.: A(x) = x + tan(A(x)^2).

Original entry on oeis.org

1, 2, 12, 120, 1680, 30480, 678720, 17902080, 545529600, 18854519040, 728651911680, 31133305082880, 1457247407616000, 74151941277173760, 4075563460173004800, 240617659203765043200, 15186689706926068531200, 1020415122190724766105600, 72722026905140804154163200
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
where A(x - tan(x^2)) = x and A(x) = x + tan(A(x)^2).
Series expansions:
A(x) = x + tan(x^2) + d/dx tan(x^2)^2/2! + d^2/dx^2 tan(x^2)^3/3! + d^3/dx^3 tan(x^2)^4/4! +...
log(A(x)/x) = tan(x^2)/x + d/dx (tan(x^2)^2/x)/2! + d^2/dx^2 (tan(x^2)^3/x)/3! + d^3/dx^3 (tan(x^2)^4/x)/4! +...

Crossrefs

Cf. A226760, A215188, A226758.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Tan[x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 23 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-tan(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, tan(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, tan(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.: Series_Reversion(x - tan(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tan(x^2)^n/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*tan(x^2)^n/n! ).
a(n) ~ n^(n-1) * sqrt(r*s/(1 + 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.4749798472710374904... is the root of the equation 2*s = cos(s^2)^2, and r = s - tan(s^2) = 0.2454667961619663296... - Vaclav Kotesovec, Jan 23 2014

A226760 E.g.f.: A(x) = x + tanh(A(x)^2).

Original entry on oeis.org

1, 2, 12, 120, 1680, 30000, 651840, 16692480, 492307200, 16432899840, 612414190080, 25205181281280, 1135434972672000, 55567263749437440, 2935699697464934400, 166525788640142131200, 10094443872515806003200, 651211844046521735577600, 44547029929095149430374400
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
where A(x - tanh(x^2)) = x and A(x) = x + tanh(A(x)^2).
Series expansions:
A(x) = x + tanh(x^2) + d/dx tanh(x^2)^2/2! + d^2/dx^2 tanh(x^2)^3/3! + d^3/dx^3 tanh(x^2)^4/4! +...
log(A(x)/x) = tanh(x^2)/x + d/dx (tanh(x^2)^2/x)/2! + d^2/dx^2 (tanh(x^2)^3/x)/3! + d^3/dx^3 (tanh(x^2)^4/x)/4! +...

Crossrefs

Cf. A226759, A215188, A226758.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Tanh[x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 23 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-tanh(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, tanh(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, tanh(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.: Series_Reversion(x - tanh(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) tanh(x^2)^n/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*tanh(x^2)^n/n! ).
a(n) ~ n^(n-1) * sqrt(r*s/(1 - 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.5456469378931069437... is the root of the equation 2*s = cosh(s^2)^2, and r = s - tanh(s^2) = 0.2564125251556591672... - Vaclav Kotesovec, Jan 23 2014

A236357 E.g.f. satisfies: sin(x + A(x)) = A(x)^2.

Original entry on oeis.org

-1, 2, -12, 120, -1680, 30360, -672000, 17599680, -532224000, 18248902560, -699544581120, 29645099003520, -1376154991411200, 69445751181206400, -3785191236642816000, 221613328121888102400, -13870531255760776396800, 924194452386724174656000
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 23 2014

Keywords

Comments

For n < 10 is |a(n)| = A215188(n), but a(10) is different.

Crossrefs

Cf. A215188, A143135.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[ArcSin[x^2]-x,{x,0,20}],x],x] * Range[0,20]!]

Formula

E.g.f.: Series_Reversion(arcsin(x^2)-x).
a(n) ~ (-1)^n * (4/sqrt(5)-8/5)^(1/4) * n^(n-1) / (exp(n) * (sqrt(sqrt(5)-2) - arcsin(sqrt(5)-2))^(n-1/2))
Showing 1-4 of 4 results.

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