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.

A227464 E.g.f. equals the series reversion of sin(x) / exp(x).

Original entry on oeis.org

1, 2, 10, 80, 884, 12480, 214600, 4352000, 101696400, 2690754560, 79516330400, 2595903897600, 92782304200000, 3603511009280000, 151115361757776000, 6805240665866240000, 327547876406050976000, 16780408888535285760000, 911669878205463707200000
Offset: 1

Views

Author

Paul D. Hanna, Jul 14 2013

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 884*x^5/5! + 12480*x^6/6! +...
where A( sin(x)/exp(x) ) = x.

Crossrefs

Cf. A227463.

Programs

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

Formula

E.g.f. A(x) satisfies: A(x) = asin(x*exp(A(x))).
a(n) ~ 2^((n-1)/2) * exp(n*Pi/4) * n^(n-1) / exp(n). - Vaclav Kotesovec, Jan 10 2014

A334856 E.g.f. A(x) satisfies: A(x) = sin(x * exp(A(x))).

Original entry on oeis.org

1, 2, 8, 48, 376, 3552, 38072, 433664, 4677184, 29762560, -793410432, -54950522880, -2386117894016, -93904016252928, -3620999060747776, -140559707495727104, -5551125947636719616, -223832352597495644160, -9212210790951194357760, -385841470335205512314880
Offset: 1

Views

Author

Ilya Gutkovskiy, May 13 2020

Keywords

Crossrefs

Cf. A227463, A227465.

Programs

  • Mathematica
    nmax = 20; CoefficientList[InverseSeries[Series[ArcSin[x]/Exp[x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest

Formula

E.g.f.: series reversion of arcsin(x) / exp(x).

A227462 E.g.f. equals the series reversion of x - x^2*exp(x^2).

Original entry on oeis.org

1, 2, 12, 144, 2400, 50760, 1310400, 39984000, 1407490560, 56143130400, 2502730137600, 123302540148480, 6653043673436160, 390182090889951360, 24713127865269043200, 1681180295452142284800, 122252619549518954496000, 9463442426016057083404800, 776944497581062575154790400
Offset: 1

Views

Author

Paul D. Hanna, Jul 13 2013

Keywords

Comments

Limit n->infinity (a(n)/n!)^(1/n) = Limit n->infinity a(n+1)/a(n)/n = 4.6910715... = 2*(1+s^2)/(s*(1+2*s^2)), where s is defined below. - Vaclav Kotesovec, added Jul 28 2013, updated Jan 07 2014

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 2400*x^5/5! +...
where A(x) = x + A(x)^2*exp(A(x)^2).
The e.g.f. satisfies:
(3) A(x) = x + x^2*exp(A(x)^2) + 2*x^3*exp(2*A(x)^2) + 5*x^4*exp(3*A(x)^2) + 14*x^5*exp(4*A(x)^2) + 42*x^6*exp(5*A(x)^2) +...
(4) log(A(x)/x) = x*exp(A(x)^2) + 3*x^2*exp(2*A(x)^2)/2 + 10*x^3*exp(3*A(x)^2)/3 + 35*x^4*exp(4*A(x)^2)/4 + 126*x^5*exp(5*A(x)^2)/5 +...
(5) A(x) = x + x^2/2*exp(x^2) + d/dx x^4/4*exp(2*x^2)/2! + d^2/dx^2 x^6/8*exp(3*x^2)/3! + d^3/dx^3 x^8/16*exp(4*x^2)/4! +...
(6) log(A(x)/x) = x*exp(x^2)/2 + d/dx x*exp(2*x^2)/2! + d^2/dx^2 x^2*exp(3*x^2)/3! + d^3/dx^3 x^3*exp(4*x^2)/4! +...

Crossrefs

Cf. A213643, A227463, A000108.

Programs

  • Mathematica
    Table[(n-1)!*SeriesCoefficient[(x/(x-x^2*E^(x^2)))^n,{x,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Jul 28 2013 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x^2*exp(x^2 +x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    /* E.g.f. A(x) = x*Catalan( x*exp(A(x)^2) ): */
{a(n)=local(A=x); for(i=1,n,A=(1-sqrt(1-4*x*exp(A^2 +x^2*O(x^n)) ))/2*exp(-A^2 +x*O(x^n)) ); 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); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*exp(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, x^(2*m-1)*exp(x^2+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = x + A(x)^2*exp(A(x)^2).
(2) A(x) = x*Catalan( x*exp(A(x)^2) ) where Catalan(x) = (1-sqrt(1-4*x))/(2*x).
(3) A(x) = x*Sum_{n>=0} binomial(2*n+1,n)/(2*n+1) * x^n * exp(n*A(x)^2).
(4) A(x) = x*exp( Sum_{n>=1} binomial(2*n-1,n) * x^n/n * exp(n*A(x)^2) ).
(5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * exp(n*x^2) / n!.
(6) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * exp(n*x^2) / n! ).
a(n) ~ n^(n-1) * s * sqrt((1+2*s^2)/(2+10*s^2+4*s^4)) / (exp(n) * ((s*(1+2*s^2))/(2*(1+s^2)))^n), where s = 0.3788063540000847107637564... is the root of the equation 2*s*(1+s^2)*exp(s^2) = 1. - Vaclav Kotesovec, Jan 07 2014

A334855 E.g.f. A(x) satisfies: A(x) = tanh(x * exp(A(x))).

Original entry on oeis.org

1, 2, 7, 32, 141, -48, -20509, -572416, -13236455, -290711040, -6196120193, -125279895552, -2200506714587, -21844509804544, 731664799971163, 69257460487553024, 3772508483052129329, 174474102931077464064, 7365599642083503837175, 286164434343334028247040
Offset: 1

Views

Author

Ilya Gutkovskiy, May 13 2020

Keywords

Crossrefs

Cf. A227463, A227465.

Programs

  • Mathematica
    nmax = 20; CoefficientList[InverseSeries[Series[ArcTanh[x]/Exp[x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest

Formula

E.g.f.: series reversion of arctanh(x) / exp(x).
Showing 1-4 of 4 results.

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