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.

A143136 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2.

Original entry on oeis.org

1, 2, 12, 128, 1920, 36992, 870912, 24232448, 777999360, 28309164032, 1151292628992, 51750540443648, 2547747292446720, 136336755956252672, 7879446478581399552, 489119124160488931328, 32456290094449950720000
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2008

Keywords

Comments

Radius of convergence is r = log(sqrt(2)+1)/2 - (sqrt(2)-1)/2 = 0.2335800...,
where A(r) = log(1+sqrt(2))/2 = arcsinh(1)/2 = 0.44068679...

Examples

			A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
sinh(A(x)) = G(x) is the e.g.f. of A143137:
G(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! + ...
Related expansions:
A(x) = x + sinh(x)^2 + d/dx sinh(x)^4/2! + d^2/dx^2 sinh(x)^6/3! + d^3/dx^3 sinh(x)^8/4! + ...
log(A(x)/x) = sinh(x)^2/x + d/dx (sinh(x)^4/x)/2! + d^2/dx^2 (sinh(x)^6/x)/3! + d^3/dx^3 (sinh(x)^8/x)/4! +...

Links

Crossrefs

Cf. A143134, A143137, A213643.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Sinh[x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-sinh(x+x*O(x^n))^2),n)}
  • PARI
    {a(n)=local(A=x);for(i=0,n,A=x + sinh(A)^2);n!*polcoeff(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, sinh(x+x*O(x^n))^(2*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, sinh(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f.: A(x) = Series_Reversion( x - sinh(x)^2 ).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sinh(x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = 1/(1 - sinh(2*A(x))).
a(n) ~ 2^(n-5/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A143135 E.g.f. satisfies: A(x) = sin(x + A(x)^2) with A(0)=0.

Original entry on oeis.org

1, 2, 11, 100, 1261, 20342, 399671, 9256840, 246907321, 7452534122, 251099460611, 9341422237420, 380293239870181, 16815919738248542, 802553031266952431, 41117164304824602640, 2250747364089063475441
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2008

Keywords

Comments

Radius of convergence of A(x) is r = Pi/4 - 1/2, with A(r) = sqrt(2)/2.

Examples

			A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134:
G(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...

Links

Crossrefs

Cf. A143134, A143137.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSin[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 19 2014 *)
  • PARI
    {a(n)=local(A=x);for(i=0,n,A=x + sin(A)^2);n!*polcoeff(sin(A),n)}
  • PARI
    {a(n)=n!*polcoeff(sin(serreverse(x-sin(x+x*O(x^n))^2)),n)}

Formula

E.g.f.: A(x) = sin(G(x)) where G(x) = x + A(x)^2 is the e.g.f. of A143134.
E.g.f. derivative: A'(x) = sqrt(1 - A(x)^2)/(1 - 2*A(x)*sqrt(1 - A(x)^2)).
a(n) ~ GAMMA(1/3) * 4^(n-1) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - Vaclav Kotesovec, Jan 19 2014

A215094 E.g.f. satisfies A(x) = sinh(x + A(x)^2/2).

Original entry on oeis.org

1, 1, 4, 25, 211, 2296, 30619, 482455, 8768596, 180603511, 4157281129, 105764735440, 2946911156281, 89247262497121, 2919028298593684, 102543779766289705, 3850690682004992491, 153927330069247143976, 6525942204725963508259, 292483420180063453725175
Offset: 1

Views

Author

Paul D. Hanna, Aug 02 2012

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 211*x^5/5! + 2296*x^6/6! +...
A(x) = sinh(G(x)) where G(x) is the e.g.f. of A215093:
G(x) = x + x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...
where
A(x)^2/2 = x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! +...

Crossrefs

Cf. A215093, A143137.

Programs

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

Formula

E.g.f.: A(x) = sinh(G(x)) where G(x) = Series_Reversion(x - sinh(x)^2) is the e.g.f. of A215093.
a(n) ~ 2^(2*n-3/2) * sqrt(1+1/sqrt(5)) * n^(n-1) / (exp(n) * (1-sqrt(5) + 4*arcsinh(sqrt((sqrt(5)-1)/2)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014
Showing 1-3 of 3 results.

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