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.

Previous Showing 11-14 of 14 results.

A280570 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

Original entry on oeis.org

1, 4, 28, 332, 5748, 131940, 3791692, 131375324, 5343640212, 250142552212, 13271217848604, 788346022938556, 51916178572447140, 3759254932421361284, 297243198474965188732, 25513664852425377663756, 2365246919693613357168916, 235776253411115081902083556, 25174157913006507920211300588, 2869108641038261410331666767772
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 28*x^3/3! + 332*x^4/4! + 5748*x^5/5! + 131940*x^6/6! + 3791692*x^7/7! + 131375324*x^8/8! + 5343640212*x^9/9! + 250142552212*x^10/10! + 13271217848604*x^11/11! + 788346022938556*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 3*G(x) where
G(x) = x^2/2! + 4*x^3/3! + 28*x^4/4! + 332*x^5/5! + 5748*x^6/6! + 131940*x^7/7! + 3791692*x^8/8! + 131375324*x^9/9! + 5343640212*x^10/10! + 250142552212*x^11/11! + 13271217848604*x^12/12! +...
Also, A(x) = x + 4 * G( (A(x) + 3*x)/4 ).
RELATED SERIES.
We have (A(x) + 3*x)/4 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 3*x)/4 = x + x^2/2! + 7*x^3/3! + 83*x^4/4! + 1437*x^5/5! + 32985*x^6/6! + 947923*x^7/7! + 32843831*x^8/8! + 1335910053*x^9/9! + 62535638053*x^10/10! + 3317804462151*x^11/11! + 197086505734639*x^12/12! +...
Further, A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3), which begins
A( (A(x) + 3*x)/4 ) = x + 5*x^2/2! + 47*x^3/3! + 707*x^4/4! + 14825*x^5/5! + 401033*x^6/6! + 13340739*x^7/7! + 528281555*x^8/8! + 24323141773*x^9/9! + 1279128727141*x^10/10! + 75770789421947*x^11/11! + 4999463984999615*x^12 +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280573, A280574, A280575.

Programs

  • Mathematica
    m = 21; A[_] = 0;
Do[A[x_] = -3 x + 4 InverseSeries[x - Integrate[A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x] * Range[0, m-1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=1, q=3) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 1, 3), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=1, q=3) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 1, 3), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=1; q=3; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Formula

E.g.f. A(x) satisfies:
(1) A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.
(2) A(x) = x + 4 * G( (A(x) + 3*x)/4 ), where G(x) = Integral A(x) dx.
(3) A(x) = -3*x + 4 * Series_Reversion(x - Integral A(x) dx).
(4) A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3).
(5) A'(x - Integral A(x) dx) = (1 + 3*A(x))/(1 - A(x)).
a(n) = Sum_{k=0..n} A277410(n,k) * 4^(n-k-1).

A067146 Shifts left under exponential reversion.

Original entry on oeis.org

1, 1, -1, 4, -29, 309, -4383, 78121, -1684706, 42801222, -1255919755, 41918624013, -1572257236114, 65619165625383, -3022617826829288, 152615633802149416, -8397224009015443509, 500957609480739613321
Offset: 1

Views

Author

Christian G. Bower, Jan 03 2002

Keywords

Comments

A signed version of A210949.

Links

Crossrefs

Cf. A210949, A277410.

Formula

E.g.f. satisfies A^(-1)(x) = A'(x) - 1.
a(n) = (-1)^n * Sum_{k=0..n-1} A277410(n,k), for n>1. - Paul D. Hanna, Jan 06 2017

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

A300282 E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x)))^2.

Original entry on oeis.org

1, 2, 14, 176, 3256, 80464, 2508432, 94989088, 4253855744, 220832422912, 13086217711232, 874394085363328, 65223621800358400, 5386555341558256640, 489099399479737741824, 48537534403454393734656, 5237456183532299486332928, 611754844797595116933365760, 77043077295500803834985809920, 10424760435433013614451330201600
Offset: 1

Views

Author

Paul D. Hanna, Mar 07 2018

Keywords

Examples

			G.f.: A(x) = x + 2*x^2/2! + 14*x^3/3! + 176*x^4/4! + 3256*x^5/5! + 80464*x^6/6! + 2508432*x^7/7! + 94989088*x^8/8! + 4253855744*x^9/9! + ...
such that A'(x) = 1/(1 - A(A(x)))^2.
RELATED SERIES.
(1 - A(x))^2 = 1 - 2*x - 2*x^2/2! - 16*x^3/3! - 216*x^4/4! - 4192*x^5/5! - 107376*x^6/6! - 3444384*x^7/7! - 133557568*x^8/8! - 6103032256*x^9/9! + ...
Series_Reversion(A(x)) = x - 2*x^2/2! - 2*x^3/3! - 16*x^4/4! - 216*x^5/5! - 4192*x^6/6! - 107376*x^7/7! + ...
A(A(x)) = x + 4*x^2/2! + 40*x^3/3! + 656*x^4/4! + 15152*x^5/5! + 455280*x^6/6! + 16947776*x^7/7! + 756120640*x^8/8! + 39475100864*x^9/9! + ...

Crossrefs

Cf. A210949.

Programs

  • PARI
    {a(n)=local(A=x, G); for(i=1, n, G = intformal( (1 - A +x*O(x^n))^2 ); A = serreverse(G)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=x); for(i=1, n, A = serreverse(intformal( (1-A +x*O(x^n))^2 ))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral (1 - A(x))^2 dx ).
(2) A'(x) = 1 / (1 - A(A(x)))^2.
(3) A''(x) = 2 / ( (1 - A(A(x)))^5 * (1 - A(A(A(x))))^2 ).
Previous Showing 11-14 of 14 results.

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