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.

A143134 E.g.f. satisfies: A(x) = x + sin( A(x) )^2 with A(0)=0.

Original entry on oeis.org

1, 2, 12, 112, 1440, 23552, 467712, 10926592, 293544960, 8914583552, 301957742592, 11285975498752, 461367611228160, 20477098870833152, 980591931131953152, 50393637174029320192, 2766350676943951626240
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) = Pi/4.

Examples

			A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
sin(A(x)) = G(x) is the e.g.f. of A143135:
G(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1261*x^5/5! +...
G(x)^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
Related expansions:
A(x) = x + sin(x)^2 + d/dx sin(x)^4/2! + d^2/dx^2 sin(x)^6/3! + d^3/dx^3 sin(x)^8/4! +...
log(A(x)/x) = sin(x)^2/x + d/dx (sin(x)^4/x)/2! + d^2/dx^2 (sin(x)^6/x)/3! + d^3/dx^3 (sin(x)^8/x)/4! +...

Links

Crossrefs

Cf. A143135, A143136, A214770.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Sin[x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 18 2014 *)
  • PARI
    {a(n)=local(A=x);for(i=0,n,A=x + sin(A)^2);n!*polcoeff(A,n)}
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))^2),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+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, sin(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. satisfies:
(1) A(x) = Series_Reversion( x - sin(x)^2 ).
(2) A(x) = x + Sum_{n>=1} (-1)^(n-1)*2^(2*n-1) * A(x)^(2*n)/(2*n)!.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^(2*n)/n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sin(x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = 1/(1 - 2*sqrt(A(x)-x)*sqrt(1+x-A(x))); thus A'(x) = 1/(1 - sin(2*A(x))).
Let f(x) = 1/(1-sin(2*x)). Then a(n) = (f(x)*d/dx)^(n-1) f(x) evaluated at x = 0. - Peter Bala, Oct 12 2011
a(n) ~ GAMMA(1/3) * 2^(2*n-3/2) * n^(n-5/6) / (3^(1/6) * sqrt(Pi) * exp(n) * (Pi-2)^(n-1/3)). - Vaclav Kotesovec, Jan 18 2014

A143137 E.g.f. satisfies A(x) = sinh(x + A(x)^2).

Original entry on oeis.org

1, 2, 13, 140, 2101, 40502, 954073, 26557400, 852911401, 31042592042, 1262704455013, 56767589130980, 2795116027239901, 149590982933830622, 8646295934108179633, 536766403519254357680, 35620604244949591333201
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2008

Keywords

Examples

			A(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! +...
A(x) = sinh(G(x)) where G(x) is the e.g.f. of A143136:
G(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...

Links

Crossrefs

Cf. A143135, A143136.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSinh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • PARI
    {a(n)=n!*polcoeff(sinh(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(sinh(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 A143136.
a(n) ~ sqrt(1+sqrt(2)) * 2^(n-7/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A185190 E.g.f. satisfies: A(x) = x + arcsinh(A(x))^2.

Original entry on oeis.org

1, 2, 12, 112, 1440, 23648, 473088, 11164032, 303624960, 9351301632, 321717276672, 12228424826880, 508916576243712, 23016333612318720, 1124014843389984768, 58949533609403842560, 3304473379374295744512, 197167421810210663301120, 12476358616574849161101312
Offset: 1

Views

Author

Paul D. Hanna, Feb 05 2012

Keywords

Comments

Radius of convergence of the e.g.f. A(x) is r = 0.2767308231516982...,
where r and A(r) satisfy: r = A(r) - (1 + A(r)^2)/4 and
A(r) = sinh( sqrt(1 + A(r)^2)/2 ), so that A(r) = 0.6241087588791013...

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ...
Related expansions:
arcsinh(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1269*x^5/5! + ...
arcsinh(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ...
Series expressions:
A(x) = x + arcsinh(x)^2 + d/dx arcsinh(x)^4/2! + d^2/dx^2 arcsinh(x)^6/3! + d^3/dx^3 arcsinh(x)^8/4! + ...
log(A(x)/x) = arcsinh(x)^2/x + d/dx (arcsinh(x)^4/x)/2! + d^2/dx^2 (arcsinh(x)^6/x)/3! + d^3/dx^3 (arcsinh(x)^8/x)/4! + ...

Crossrefs

Cf. A143136, A205886, A214431.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - ArcSinh[x]^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 11 2014 *)
nmax = 20; A[] = 0; Do[A[x] = x + ArcSinh[A[x]]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Aug 10 2025 *)
  • PARI
    {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + asinh(A)^2); n!*polcoeff(A, n)}
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-asinh(x+x*O(x^n))^2), n)}
for(n=1,26,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, asinh(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, asinh(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. satisfies:
(1) A(x) = Series_Reversion( x - arcsinh(x)^2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) arcsinh(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (arcsinh(x)^(2*n)/x) / n! ).
(4) A'(x) = sqrt(1+A(x)^2) / (sqrt(1+A(x)^2) - 2*arcsinh(A(x))).
a(n) ~ n^(n-1) * sqrt((1+s^2)/(2-s)) / (exp(n) * ((4*s-1-s^2)/4)^(n-1/2)), where s = 0.62410875887910134116443... is the root of the equation 2*log(s+sqrt(1+s^2)) = sqrt(1+s^2). - Vaclav Kotesovec, Jan 11 2014

A215093 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2 / 2.

Original entry on oeis.org

1, 1, 3, 19, 165, 1801, 24003, 378379, 6880485, 141757201, 3263757123, 83046239299, 2314209491685, 70093262093401, 2292753819807363, 80548997707137979, 3024937662747436965, 120925183043471954401, 5127013172890341294723, 229794790034059392232819
Offset: 1

Views

Author

Paul D. Hanna, Aug 02 2012

Keywords

Comments

Radius of convergence is r = log(sqrt(5)+2)/2 - (sqrt(5)-1)/4 = 0.41280074...,
where A(r) = log(sqrt(5)+2)/2 = arcsinh(2)/2 = 0.72181773...

Examples

			A(x) = x + x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! + ...
sinh(A(x)) = G(x) is the e.g.f. of A215094:
G(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 211*x^5/5! + 2296*x^6/6! + ...
where
G(x)^2/2 = x^2/2! + 3*x^3/3! + 19*x^4/4! + 165*x^5/5! + ...
Related expansions:
A(x) = x + sinh(x)^2/2 + d/dx sinh(x)^4/(2!*2^2) + d^2/dx^2 sinh(x)^6/(3!*2^3) + d^3/dx^3 sinh(x)^8/(4!*2^4) + ...
log(A(x)/x) = sinh(x)^2/(2*x) + d/dx sinh(x)^4/(2!*2^2*x) + d^2/dx^2 sinh(x)^6/(3!*2^3*x) + d^3/dx^3 sinh(x)^8/(4!*2^4*x) + ...

Crossrefs

Cf. A215094, A143136.

Programs

  • Mathematica
    max = 20; Rest[ CoefficientList[ InverseSeries[ Series[x - Sinh[x]^2/2, {x, 0, max}], x], x]]*Range[max]! (* Jean-François Alcover, Aug 06 2012, from 1st formula *)
  • PARI
    {a(n)=n!*polcoeff(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(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!*2^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)/(m!*2^m*x))+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. satisfies:
(1) A(x) = Series_Reversion( x - sinh(x)^2/2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/(n!*2^n).
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/(n!*2^n*x) ).
(4) A'(x) = 1/(1 - sinh(2*A(x))/2).
a(n) ~ 2^(2*n-1) * n^(n-1) / (5^(1/4) * exp(n) * (1-sqrt(5) + log(9+4*sqrt(5)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014
Showing 1-4 of 4 results.

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