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.

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

A120596 G.f. satisfies: 6*A(x) = 5 + x + A(x)^5, starting with [1,1,10].

Original entry on oeis.org

1, 1, 10, 210, 5505, 161601, 5082420, 167451780, 5705082795, 199354509755, 7105393162010, 257312347583330, 9440808323869455, 350189693739455535, 13110655796699158800, 494772468434359266960, 18801468275832345890970
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 + x + 10*x^2 + 210*x^3 + 5505*x^4 + 161601*x^5 +...
A(x)^5 = 1 + 5*x + 60*x^2 + 1260*x^3 + 33030*x^4 + 969606*x^5 +...

Crossrefs

Cf. A120588 - A120595, A120597 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[1+6*x - (1+x)^5, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+x+10*x^2+x*O(x^n));for(i=0,n,A=A-6*A+5+x+A^5);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion(1+6*x - (1+x)^5). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (5+x)^(4*n+1)/6^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-5 + 4*(6/5)^(5/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A245043 G.f. satisfies: A(x) = (12 + A(x)^4) / (13 - 27*x).

Original entry on oeis.org

1, 3, 15, 117, 1158, 12930, 154986, 1947582, 25317009, 337610451, 4592807895, 63488144109, 889226772132, 12592147132572, 179982549300948, 2593187073716460, 37622924436008574, 549178914689660106, 8059539548880228138, 118846096104074358942, 1760035125442960123992
Offset: 0

Views

Author

Paul D. Hanna, Jul 10 2014

Keywords

Examples

			G.f.: A(x) =  1 + 3*x + 15*x^2 + 117*x^3 + 1158*x^4 + 12930*x^5 +...
Compare A(x)^4 to (13-27*x)*A(x):
A(x)^4 = 1 + 12*x + 114*x^2 + 1116*x^3 + 11895*x^4 + 136824*x^5 +...
(13-27*x)*A(x) = 13 + 12*x + 114*x^2 + 1116*x^3 + 11895*x^4 + 136824*x^5 +...

Crossrefs

Cf. A120595, A245009, A244627, A244594, A244856.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/(27*(1+x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
  • PARI
    {a(n)=polcoeff(1 + serreverse( (1+13*x - (1+x)^4)/(27*(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 - (13-27*x)*Ax )/9 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = 1 + Series_Reversion( (1+13*x - (1+x)^4)/(27*(1+x)) ).
(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x*A(x))^(3*n+1) / 13^(4*n+1).
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (12+27*x + G(x)^4)/13 is the g.f. of A120595.
a(n) ~ 3^(3*n - 7/2) / (2^(7/4) * sqrt(Pi) * n^(3/2) * (13 - 8*sqrt(2))^(n - 3/2)). - Vaclav Kotesovec, Nov 27 2017
Showing 1-3 of 3 results.

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