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.

A120592 G.f. satisfies: 5*A(x) = 4 + 4*x + A(x)^3, starting with [1,2,6].

Original entry on oeis.org

1, 2, 6, 40, 330, 3048, 30156, 312528, 3349170, 36809960, 412651668, 4700098416, 54237852708, 632762593680, 7450815536280, 88435205367456, 1056940049423682, 12708927083800296, 153636691533864900, 1866178021496170800
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006, Jan 24 2008

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 2*x + 6*x^2 + 40*x^3 + 330*x^4 + 3048*x^5 + 30156*x^6 +...
A(x)^3 = 1 + 6*x + 30*x^2 + 200*x^3 + 1650*x^4 +15240*x^5 +150780*x^6 +...

Links

Crossrefs

Cf. A120588 - A120591, A120593 - A120607.

Programs

  • Mathematica
    FullSimplify[Table[SeriesCoefficient[Sum[Binomial[3*k,k]/(2*k+1)*(4+4*x)^(2*k+1)/5^(3*k+1),{k,0,Infinity}],{x,0,n}],{n,0,20}]] (* Vaclav Kotesovec, Oct 19 2012 *)
  • PARI
    {a(n)=local(A=1+2*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-5*A+4+4*x+A^3)/2);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+5*x - (1+x)^3)/4).
G.f.: A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (4+4*x)^(2*n+1) / 5^(3*n+1), due to Lagrange Inversion.
Recurrence: 17*(n-1)*n*a(n) = 108*(n-1)*(2*n-3)*a(n-1) + 12*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ sqrt(250-60*sqrt(15))*((108+30*sqrt(15))/17)^n/(30*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

A120594 G.f. satisfies: 8*A(x) = 7 + 8*x + A(x)^4, starting with [1,2,6].

Original entry on oeis.org

1, 2, 6, 44, 394, 3948, 42364, 476120, 5532714, 65935804, 801461012, 9897836520, 123840983812, 1566487308344, 19999112293944, 257365488659376, 3334967582746218, 43477505482249692, 569854228738577572
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 2*x + 6*x^2 + 44*x^3 + 394*x^4 + 3948*x^5 + 42364*x^6 +...
A(x)^4 = 1 + 8*x + 48*x^2 + 352*x^3 + 3152*x^4 + 31584*x^5 + 338912*x^6+..

Crossrefs

Cf. A120588 - A120593, A120595 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+2*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-8*A+7+8*x+A^4)/4);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+8*x - (1+x)^4)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7+8*x)^(3*n+1)/8^(4*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(-11/6 + 3*n) * (-7 + 6*2^(1/3))^(1/2 - n) / (n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Nov 28 2017

A244856 G.f. satisfies: A(x) = (4 + A(x)^4) / (5-x).

Original entry on oeis.org

1, 1, 7, 95, 1614, 30718, 626434, 13383650, 295692145, 6700461777, 154871912815, 3637093846055, 86539594779772, 2081721640140460, 50542732376144460, 1236960716959913020, 30483096737455969766, 755783491624380578998, 18839297079646725396450
Offset: 0

Views

Author

Paul D. Hanna, Jul 09 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 95*x^3 + 1614*x^4 + 30718*x^5 +...
Compare A(x)^4 to (5-x)*A(x):
A(x)^4 = 1 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...
(5-x)*A(x) = 5 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...

Crossrefs

Cf. A120593, A244627, A244594.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+5*x - (1+x)^4)/(1+x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
  • PARI
    {a(n)=polcoeff(1 + serreverse((1+5*x - (1+x)^4)/(1 + x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=[1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^4 - (5-x)*Ax ) )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = 1 + Series_Reversion( (1+5*x - (1+x)^4)/(1 + x) ).
(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4 + x*A(x))^(3*n+1) / 5^(4*n+1).
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (4+x + G(x)^4)/5 is the g.f. of A120593.
a(n) ~ 2^(n/2 - 2) * 3^(3*(n-1)/4) / (sqrt(Pi) * n^(3/2) * (5*sqrt(2)*3^(3/4) - 16)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017
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.