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.

A233335 E.g.f. A(x) satisfies: A( Integral 1/A(x) dx ) = exp(x).

Original entry on oeis.org

1, 1, 2, 7, 38, 292, 2975, 38350, 604433, 11351659, 249042701, 6283114723, 179995680530, 5794486077958, 207806806310354, 8241414107222095, 359171801820266717, 17107537203463252273, 886296777786378900077, 49732564234138336160086, 3011177123882906437153214, 196063383282648338166793297
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 292*x^5/5! + 2975*x^6/6! +...
Related expansions.
Integral 1/A(x) dx = x - x^2/2! - x^4/4! - 6*x^5/5! - 52*x^6/6! - 591*x^7/7! - 8404*x^8/8! +...
The series reversion of Integral 1/A(x) dx equals log(A(x)) and begins:
log(A(x)) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 126*x^5/5! + 1333*x^6/6! + 17895*x^7/7! + 293461*x^8/8! +...+ A214645(n)*x^n/n! +...
and equals the e.g.f. of A214645.

Links

Crossrefs

Cf. A233336, A214645 (log).

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(serreverse(intformal(1/A+x*O(x^n)))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. satisfies: A(x) = exp( Series_Reversion( Integral 1/A(x) dx ) ).
E.g.f.: exp(G(x)) where G(x) = exp(G(G(x))) is the e.g.f. of A214645.

A214654 E.g.f. A(x) satisfies: A'(x) = exp( A(x)*A'(x) ).

Original entry on oeis.org

1, 1, 4, 29, 307, 4288, 74511, 1550203, 37588412, 1041217039, 32446359005, 1123624632224, 42814687805649, 1780347364682777, 80231627759556196, 3895332991309376213, 202713074683790193475, 11256955024502873008864, 664444403260495390747071
Offset: 1

Views

Author

Paul D. Hanna, Jul 24 2012

Keywords

Comments

Compare to the trivial identity: G'(x) = exp(G(x)) when G(x) = -log(1-x).

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 29*x^4/4! + 307*x^5/5! + 4288*x^6/6! +...
By definition log(A'(x)) = A(x)*A'(x), where:
(3) A'(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 307*x^4/4! + 4288*x^5/5! +...
(4) A(x)*A'(x) = x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2437*x^5/5! + 40523*x^6/6! + 814355*x^7/7! + 19196769*x^8/8! + 519397829*x^9/9! +...
RELATED SERIES:
Let W(x) = x^2/2! + x^3/3! + 4*x^4/4! + 27*x^5/5! + 256*x^6/6! + 3125*x^7/7! +...+ (n-2)^(n-2)*x^n/n! +... then
(6) A(x) = x + W(A(x)); equivalently, A(x - W(x)) = x.
(7) A(x) = x + W(x) + d/dx W(x)^2/2! + d^2/dx^2 W(x)^3/3! + d^3/dx^3 W(x)^4/4! +...
(8) log(A(x)/x) = W(x)/x + d/dx W(x)^2/(2!*x) + d^2/dx^2 W(x)^3/(3!*x) + d^3/dx^3 W(x)^4/(4!*x) +...

Links

Crossrefs

Cf. A210949, A214645.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[-((x^2*(1 + 2*LambertW[-x])) / (4*LambertW[-x]^2)) + 1/4,{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Feb 21 2014 *)
  • PARI
    {a(n)=local(A=x);for(i=1,n,A=intformal(exp(A*A'+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,W=sum(m=2,n+1,(m-2)^(m-2)*x^m/m!)+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, W^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,W=sum(m=2,n+1,(m-2)^(m-2)*x^m/m!)+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, W^m/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A''(x) = A'(x)^3/(1 - A(x)*A'(x)).
(2) A(x) = x + Sum_{n>=2} (n-2)^(n-2)*A(x)^n/n!.
(3) A'(x) = Sum_{n>=0} (n+1)^(n-1)*A(x)^n/n!.
(4) A(x)*A'(x) = Sum_{n>=1} n^(n-1)*A(x)^n/n!.
(5) A(x) = Series_Reversion( Integral -x/LambertW(-x) dx ).
Let W(x) = Sum_{n>=2} (n-2)^(n-2)*x^n/n!, then e.g.f. A(x) satisfies:
(6) A(x) = Series_Reversion(x - W(x)).
(7) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/n!.
(8) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) W(x)^n/(n!*x) ).
a(n) ~ 2^(2*n-3) * n^(n-2) / (exp(n-2) * (1+exp(-2))^(n-3/2)). - Vaclav Kotesovec, Feb 17 2014
A(x) = Series_Reversion( -((x^2*(1 + 2*LambertW(-x))) / (4*LambertW(-x)^2)) + 1/4). - Vaclav Kotesovec, Feb 21 2014

Extensions

Program in Mathematica improved by Vaclav Kotesovec, Feb 21 2014

A259267 E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).

Original entry on oeis.org

1, 2, 12, 128, 2016, 42656, 1145280, 37563008, 1464675840, 66533778944, 3466031815680, 204489094565888, 13524452573872128, 994257291909816320, 80668058806271016960, 7179145234347383128064, 697131195162680465817600, 73522035747248454761578496, 8387016414085244676889116672
Offset: 1

Views

Author

Paul D. Hanna, Jul 03 2015

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! +...
Related expansions:
A'(x) =  1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2016*x^4/4! + 42656*x^5/5! +...
A(A(x)) = log(A'(x))/2 = x + 4*x^2/2! + 36*x^3/3! + 520*x^4/4! + 10512*x^5/5! + 276064*x^6/6! + 8987712*x^7/7! + 351278080*x^8/8! +...
The exponential of e.g.f. A(x) equals the e.g.f. of A233336:
exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! + 1677883*x^7/7! +...+ A233336(n)*x^n/n! +...

Crossrefs

Cf. A233336, A214645.

Programs

  • PARI
    {a(n)=local(A=x+x^2); for(i=0, n, A=intformal(exp(2*subst(A, x, A+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) = 2*exp( 4*A(A(x)) + 2*A(A(A(x))) ).
(2) exp(-2*A(x)) = d/dx Series_Reversion(A(x)).
(3) A(x) = log(F(x)) where F(x) satisfies: F( Integral 1/F(x)^2 dx ) = exp(x) and equals the e.g.f. of A233336.
a(n) = 2^(n-1) * A214645(n) for n>=1.

A259822 E.g.f. A(x) satisfies: A( Integral 1/A(x)^3 dx ) = exp(x).

Original entry on oeis.org

1, 1, 4, 37, 586, 13612, 424621, 16827976, 815866699, 47093387797, 3170897237125, 245127016240321, 21482473673228266, 2112385883734692910, 231062843227493844112, 27913223028923592662539, 3701041353685453743060265, 535729316331363978105167557, 84263588534262286958390813305
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2015

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 586*x^4/4! + 13612*x^5/5! + 424621*x^6/6! +...
where log(A(x)) = Series_Reversion( Integral 1/A(x)^3 dx ):
log(A(x)) = x + 3*x^2/2! + 27*x^3/3! + 432*x^4/4! + 10206*x^5/5! + 323919*x^6/6! +...+ 3^(n-1)*A214645(n)*x^n/n! +...
and
A(x/3)^3 = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 292*x^5/5! + 2975*x^6/6! +...+ A233335(n)*x^n/n! +...

Crossrefs

Cf. A233335, A214645.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(serreverse(intformal(1/A^3+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. satisfies: A(x) = exp( Series_Reversion( Integral 1/A(x)^3 dx ) ).
E.g.f. A(x) such that A(x/3)^3 is the e.g.f. of A233335.
Showing 1-4 of 4 results.

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