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 1-5 of 5 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

A214431 E.g.f. satisfies: A(x) = x + arcsin( A(x) )^2.

Original entry on oeis.org

1, 2, 12, 128, 1920, 37088, 876288, 24478848, 789208320, 28841682432, 1178155579392, 53196745543680, 2630883967315968, 141432764970270720, 8211780541909352448, 512119065654535618560, 34141138126195739394048, 2422954554718763388764160, 182378340941780059828518912
Offset: 1

Views

Author

Paul D. Hanna, Jul 17 2012

Keywords

Comments

Radius of convergence of A(x) is r = A(r) - (1-A(r)^2)/4 = 0.232465575..., where A(r) = 0.435130859... satisfies A(r) = sin(sqrt(1-A(r)^2)/2).

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
where
arcsin(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2109*x^5/5! + 40838*x^6/6! + 966393*x^7/7! + ...
arcsin(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + 37088*x^6/6! + 876288*x^7/7! + ...
Related expansions:
A(x) = x + arcsin(x)^2 + d/dx arcsin(x)^4/2! + d^2/dx^2 arcsin(x)^6/3! + d^3/dx^3 arcsin(x)^8/4! + ...
log(A(x)/x) = arcsin(x)^2/x + d/dx (arcsin(x)^4/x)/2! + d^2/dx^2 (arcsin(x)^6/x)/3! + d^3/dx^3 (arcsin(x)^8/x)/4! + ...

Crossrefs

Cf. A143134, A185190.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - ArcSin[x]^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-asin(x+x*O(x^n))^2), n)}
  • PARI
    {a(n)=local(A=x); for(i=0, n, A=x + asin(A+x*O(x^n))^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, asin(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, asin(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 - arcsin(x)^2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) arcsin(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (arcsin(x)^(2*n)/x) / n! ).
(4) A'(x) = sqrt(1-A(x)^2) / (sqrt(1-A(x)^2) - 2*arcsin(A(x))).
a(n) ~ sqrt((1-s^2)/(2+s)) * n^(n-1) / (exp(n) * r^(n-1/2)), where r and s=A(r) were described above. - Vaclav Kotesovec, Jan 12 2014

A214770 E.g.f. satisfies: A(x) = x + sin(A(x))*sinh(A(x)).

Original entry on oeis.org

1, 2, 12, 120, 1680, 30232, 664832, 17277120, 518031360, 17602865312, 668505311232, 28059791760000, 1289932186583040, 64455076284318592, 3478305412257677312, 201608948937441269760, 12491465252403224248320, 823886511479340063068672, 57633367371058675735068672
Offset: 1

Views

Author

Paul D. Hanna, Jul 31 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/5! + 30232*x^6/6! +...
where A(x - sin(x)*sinh(x)) = x and A(x) = x + sin(A(x))*sinh(A(x)).
Related expansions:
sin(x)*sinh(x) = 2*x^2/2! - 8*x^6/6! + 32*x^10/10! - 128*x^14/14! + 512*x^18/18! -+...+ (-1)^(n-1)*2^(2*n-1)*x^(4*n-2)/(4*n-2)! +-...
sin(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 108*x^4/4! + 1501*x^5/5! + 26902*x^6/6! +...
sinh(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 132*x^4/4! + 1861*x^5/5! + 33622*x^6/6! +...
Other series:
A(x) = x + sin(x)*sinh(x) + d/dx sin(x)^2*sinh(x)^2/2! + d^2/dx^2 sin(x)^3*sinh(x)^3/3! + d^3/dx^3 sin(x)^4*sinh(x)^4/4! +...
log(A(x)/x) = sin(x)*sinh(x)/x + d/dx sin(x)^2*sinh(x)^2/x/2! + d^2/dx^2 sin(x)^3*sinh(x)^3/x/3! + d^3/dx^3 sin(x)^4*sinh(x)^4/x/4! +...

Crossrefs

Cf. A143134.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - Sin[x]*Sinh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))*sinh(x+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, sin(x+x*O(x^n))^m*sinh(x+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, sin(x+x*O(x^n))^m*sinh(x+x*O(x^n))^m/x/m!))); n!*polcoeff(A, n)}

Formula

E.g.f. satisfies:
(1) A(x) = Series_Reversion(x - sin(x)*sinh(x)).
(2) A(x) = x + Sum_{n>=1} (-1)^(n-1)*2^(2*n-1) * A(x)^(4*n-2)/(4*n-2)!.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^n*sinh(x)^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^n*sinh(x)^n/x / n! ).
a(n) ~ n^(n-1) / (exp(n) * r^(n-1/2) * sqrt(2*cos(s)*cosh(s))), where s = 0.50105258964301589... is the root of the equation cosh(s)*sin(s) + cos(s)*sinh(s) = 1, and r = s - sin(s)*sinh(s) = 0.25017469884019539.... - Vaclav Kotesovec, Jan 13 2014

A277612 E.g.f. satisfies: A(x - sin(x)^2) = x + sin(x)^2.

Original entry on oeis.org

1, 4, 24, 224, 2880, 47104, 935424, 21853184, 587089920, 17829167104, 603915485184, 22571950997504, 922735222456320, 40954197741666304, 1961183862263906304, 100787274348058640384, 5532701353887903252480, 323102311113161602760704, 20000832981651983154806784, 1308180577070098190616756224, 90146906116103034082689024000, 6527896185206802934447948693504
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2016

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47104*x^6/6! + 935424*x^7/7! + 21853184*x^8/8! + 587089920*x^9/9! + 17829167104*x^10/10! +...
such that A(x - sin(x)^2) = x + sin(x)^2.
RELATED SERIES.
A(x - sin(x)^2) = x + 2*x^2/2! - 8*x^4/4! + 32*x^6/6! - 128*x^8/8! + 512*x^10/10! - 2048*x^12/12! +...
which equals x + sin(x)^2.
cos(A(x) + x) = 1 - 4*x^2/2! - 24*x^3/3! - 224*x^4/4! - 2880*x^5/5! - 47104*x^6/6! +...
which equals 1+x - A(x).

Crossrefs

Cf. A143134.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,21,A = subst(x + sin(x +x*O(x^n) )^2,x, serreverse(x - sin(x +x*O(x^n) )^2))); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {a(n) = my(A=x); for(i=0,n, A = 1+x - cos(A + x +x*O(x^n))); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = 1+x - cos(A(x) + x).
(2) A(x) = x + 2 * sin( (A(x) + x)/2 )^2.
(3) A(x) = -x + 2 * Series_Reversion(x - sin(x)^2).
a(n) = 2 * A143134(n) for n>1.
Showing 1-5 of 5 results.

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