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 1-2 of 2 results.

A120595 G.f. satisfies: 13*A(x) = 12 + 27*x + A(x)^4, starting with [1,3,6].

Original entry on oeis.org

1, 3, 6, 36, 249, 1932, 16044, 139500, 1253934, 11558316, 108658902, 1037800920, 10041891132, 98230257636, 969814634424, 9651213968784, 96710160474513, 974967422602428, 9881687141571732, 100632995795535588
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 + 3*x + 6*x^2 + 36*x^3 + 249*x^4 + 1932*x^5 +...
A(x)^4 = 1 + 12*x + 78*x^2 + 468*x^3 + 3237*x^4 + 25116*x^5 +...

Crossrefs

Cf. A120588 - A120594, A120596 - A120607; A245043.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-13*A+12+27*x+A^4)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^4)/27).
G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x)^(3*n+1) / 13^(4*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(3*n - 7/3) * 3^(2*n) / (13^(1/3) * sqrt(Pi) * n^(3/2) * (2^(1/3)*13^(4/3) - 32)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017

A120597 G.f. satisfies: 9*A(x) = 8 + 8*x + A(x)^5, starting with [1,2,10].

Original entry on oeis.org

1, 2, 10, 120, 1770, 29208, 516180, 9554640, 182867970, 3589443160, 71861735660, 1461730482160, 30123451315620, 627598216410480, 13197173403868200, 279728425129963680, 5970277970921643570, 128199003794219752920
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 + 10*x^2 + 120*x^3 + 1770*x^4 + 29208*x^5 +...
A(x)^5 = 1 + 10*x + 90*x^2 + 1080*x^3 + 15930*x^4 + 262872*x^5 +...

Crossrefs

Cf. A120588 - A120596, A120598 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+9*x - (1+x)^5)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+2*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-9*A+8+8*x+A^5)/4);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+9*x - (1+x)^5)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (8+8*x)^(4*n+1)/9^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-1 + 9*sqrt(3)/(10*5^(1/4)))^(1/2 - n) / (3^(3/4) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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