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.

A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

Original entry on oeis.org

1, 1, 5, 65, 1405, 42505, 1653125, 78578225, 4414067725, 286099718425, 21015972365525, 1725374840578625, 156560122048892125, 15559151967183795625, 1680744724811088153125, 196083244062052339084625, 24570430118524659881918125, 3291153805391398126661325625
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1405*x^4/4! + 42505*x^5/5! +...
where A(5*log(1+x) - 4*x) = 1+x.
Related series:
A(x)^5 = 1 + 5*x + 45*x^2/2! + 685*x^3/3! + 15645*x^4/4! + 485645*x^5/5! +...
1/A(x)^5 = 1 - 5*x + 5*x^2/2! - 85*x^3/3! - 1595*x^4/4! - 50645*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^5 dx))^(1/5) begins:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 +...+ A002294(n)*x^n +...
where G(x) = 1 + x*G(x)^5.

Crossrefs

Cf. A006351, A058562, A234294.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+A^5*intformal(1/(A^5+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(5*log(1+X) - 4*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-4*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1 + Series_Reversion( 5*log(1+x) - 4*x ).
E.g.f.: -5/4*LambertW(-4/5*exp((x-4)/5)).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^5 dx ) )^(1/5), where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
O.g.f.: 1 + x/(1-x - 4*x/(1-2*x - 4*2*x/(1-3*x - 4*3*x/(1-4*x - 4*4*x/(1-...))))), a continued fraction.
a(n) ~ sqrt(5) * n^(n-1) / (4*exp(n)*(5*log(5)-10*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

A234292 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x) dx.

Original entry on oeis.org

1, 1, 7, 103, 2365, 74305, 2970415, 144324775, 8259853525, 544284565825, 40589062689175, 3379489549611175, 310764664533047725, 31280478682100673025, 3420865044191897677375, 403897481132212723030375, 51205807000808636432381125, 6937963969329514181526090625
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 103*x^3/3! + 2365*x^4/4! + 74305*x^5/5! +...
where A( x*(6-3*x-x^2)/(6*(1+x)^3) ) = 1+x.
Related series:
A(x)^4 = 1 + 4*x + 40*x^2/2! + 688*x^3/3! + 17200*x^4/4! + 569680*x^5/5! +...
1/A(x) = 1 - x - 5*x^2/2! - 67*x^3/3! - 1475*x^4/4! - 45265*x^5/5! +...

Crossrefs

Cf. A234293, A234294.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[x*(6-3*x-x^2) / (6*(1+x)^3), {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)
FullSimplify[CoefficientList[Series[(Sqrt[3]*Cos[(1/6)*ArcCos[5/27 + (64*x)/9]] - 3*Sin[(1/3)*ArcSin[(1/3)*Sqrt[11/3 - 32*x]]]) / Sqrt[1+6*x], {x, 0, 10}], x] * Range[0, 10]!] (* Vaclav Kotesovec, Dec 26 2013 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^1+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=1+serreverse(x*(6 - 3*x - x^2)/(6*(1+x+x*O(x^n))^3)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1 + Series_Reversion( x*(6 - 3*x - x^2) / (6*(1+x)^3) ).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x) dx ) ), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
From Vaclav Kotesovec, Dec 26 2013: (Start)
E.g.f.: (sqrt(3)*cos(arccos(5/27 + 64*x/9)/6) - 3*sin(arcsin(sqrt(11/3 - 32*x)/3)/3)) / sqrt(1+6*x)
Recurrence: 11*a(n) = 3*(10*n-37)*a(n-1) + 64*(3*n-5)*(3*n-4)*a(n-2)
a(n) ~ 2^(5*n+3/2) * 3^(n-3) * n^(n+1/2) / (11^(n-1/2) * exp(n) * n^(3/2))
(End)

A234293 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^3 dx.

Original entry on oeis.org

1, 1, 5, 57, 1053, 27057, 891765, 35883369, 1705399821, 93486709089, 5806695485925, 403035074738073, 30915209058366717, 2597006055407919633, 237114260894842246485, 23379981536098259658441, 2475981772053312305191149, 280285314108547303508788161, 33774891856931450901116286405
Offset: 0

Views

Author

Paul D. Hanna, Dec 25 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 57*x^3/3! + 1053*x^4/4! + 27057*x^5/5! +...
where A(4*x/(1+x) - 3*log(1+x)) = 1+x.
Related series:
A(x)^3 = 1 + 3*x + 21*x^2/2! + 267*x^3/3! + 5157*x^4/4! + 135531*x^5/5! +...
A(x)^4 = 1 + 4*x + 32*x^2/2! + 432*x^3/3! + 8592*x^4/4! + 229488*x^5/5! +...
1/A(x)^3 = 1 - 3*x - 3*x^2/2! - 51*x^3/3! - 963*x^4/4! - 25011*x^5/5! +...
The series reversion of A(x)-1 begins:
4*x/(1+x) - 3*log(1+x) = x - 5*x^2/2 + 9*x^3/3 - 13*x^4/4 + 17*x^5/5 - 21*x^6/6 +...

Crossrefs

Cf. A234292, A234294, A234295.

Programs

  • Maple
    seq(n! * coeff(series(-4/(3*LambertW(-1, -4/3*exp((x-4)/3))), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 26 2013
  • Mathematica
    CoefficientList[1 + InverseSeries[Series[4*x/(1+x)-3*Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^3+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(4*X/(1+X)-3*log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1 + Series_Reversion( 4*x/(1+x) - 3*log(1+x) ).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^3 dx ) )^(1/3), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ n! * (2/3)^(3/2) / (sqrt(Pi) * n^(3/2) * (1-6*log(2)+3*log(3))^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013
E.g.f.: -4/(3*LambertW(-1,-4/3*exp((x-4)/3))). - Vaclav Kotesovec, Dec 26 2013
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.