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 1-5 of 5 results.

A120590 G.f. satisfies: 4*A(x) = 3 + x + A(x)^3, starting with [1,1,3].

Original entry on oeis.org

1, 1, 3, 19, 150, 1326, 12558, 124590, 1278189, 13449205, 144342627, 1573990275, 17389407984, 194228357568, 2189610888840, 24881753664840, 284708154606318, 3277578288381318, 37934510719585350, 441152315040444150
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 + x + 3*x^2 + 19*x^3 + 150*x^4 + 1326*x^5 + 12558*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 76*x^3 + 600*x^4 + 5304*x^5 + 50232*x^6 +...

Links

Crossrefs

Cf. A120591 (A(x)^3); A244594, A120588, A120592 - A120607.

Programs

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

Formula

G.f.: A(x) = 1 + Series_Reversion(1+4*x - (1+x)^3).
G.f.: A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (3+x)^(2*n+1) / 4^(3*n+1), due to Lagrange Inversion.
Recurrence: 13*(n-1)*n*a(n) = 81*(n-1)*(2*n-3)*a(n-1) + 3*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ sqrt(32-18*sqrt(3))*((81+48*sqrt(3))/13)^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
G.f.: 4 * sin( arcsin(3 * sqrt(3) * (3 + x) / 16) / 3) / sqrt(3). - Benedict W. J. Irwin, Oct 19 2016

A244627 G.f. satisfies: A(x) = (4 + A(x)^3) / (5 - 4*x).

Original entry on oeis.org

1, 2, 10, 84, 882, 10380, 130916, 1729960, 23640770, 331357276, 4737405356, 68818101400, 1012852747220, 15070913484664, 226340757825800, 3426481380787024, 52232521742541410, 801068351764540540, 12351730365539402076, 191363830520437179960
Offset: 0

Views

Author

Paul D. Hanna, Jul 02 2014

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 84*x^3 + 882*x^4 + 10380*x^5 + 130916*x^6 +...
Compare
(5 - 4*x)*A(x) = 5 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 +...
to:
A(x)^3 = 1 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 + 613060*x^6 +...

Links

Crossrefs

Cf. A120592, A244594.

Programs

  • PARI
    {a(n)=polcoeff(1 + 2*serreverse(x*(1 - 3*x - 2*x^2)/(1 + 2*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^3 - (5-4*x)*Ax )/2 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = 1 + 2*Series_Reversion( x*(1 - 3*x - 2*x^2)/(1 + 2*x) ).
(2) A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (4 + 4*x*A(x))^(2*n+1) / 5^(3*n+1).
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (3+x + G(x)^3)/4 is the g.f. of A120592.
Recurrence: 17*(n-1)*n*a(n) = 150*(n-1)*(2*n-3)*a(n-1) - 40*(6*n^2 - 24*n + 23)*a(n-2) + 32*(n-2)*(2*n-7)*a(n-3). - Vaclav Kotesovec, Jul 03 2014
a(n) ~ sqrt(15-9*2^(2/3)) * ((60*2^(2/3)+72*2^(1/3)+100)/17)^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2014

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

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

A245009 G.f. satisfies: A(x) = (7 + A(x)^4) / (8 - 8*x).

Original entry on oeis.org

1, 2, 10, 88, 978, 12200, 163156, 2286448, 33138874, 492657384, 7470940300, 115115319376, 1797128902132, 28364816229008, 451870965523368, 7256283996155360, 117333885356923274, 1908844190372949224, 31221135850863938268, 513100005743085437328, 8468653781083527106012, 140314257925457275837488
Offset: 0

Views

Author

Paul D. Hanna, Jul 09 2014

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 88*x^3 + 978*x^4 + 12200*x^5 +...
Compare A(x)^4 to 8*(1-x)*A(x):
A(x)^4 = 1 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...
8*(1-x)*A(x) = 8 + 8*x + 64*x^2 + 624*x^3 + 7120*x^4 + 89776*x^5 +...

Crossrefs

Cf. A120594, A244627, A244594, A244856, A245043.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/(8*(1+x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
  • PARI
    {a(n)=polcoeff(1 + serreverse((1+8*x - (1+x)^4)/(8*(1+x +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=[1], Ax=1+2*x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^4 - 8*(1-x)*Ax )/4 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = 1 + Series_Reversion( (1+8*x - (1+x)^4)/(8*(1+x)) ).
(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7 + 8*x*A(x))^(3*n+1) / 8^(4*n+1).
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (7+8*x + G(x)^4)/8 is the g.f. of A120594.
a(n) ~ 3^(3*(n-1)/4) * 7^((n-1)/4) / (sqrt(Pi) * n^(3/2) * (3^(3/4)*7^(1/4) - 7/2)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017
Showing 1-5 of 5 results.

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