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.

A120599 G.f. satisfies: 13*A(x) = 12 + 32*x + A(x)^5, starting with [1,4,20].

Original entry on oeis.org

1, 4, 20, 280, 4660, 86728, 1727880, 36047280, 777470580, 17195957480, 387906427480, 8890184148560, 206419640698440, 4845319424269520, 114791477960006800, 2741248077305459040, 65915164046356799220
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 + 4*x + 20*x^2 + 280*x^3 + 4660*x^4 + 86728*x^5 +...
A(x)^5 = 1 + 20*x + 260*x^2 + 3640*x^3 + 60580*x^4 + 1127464*x^5 +...

Crossrefs

Cf. A120588 - A120598, A120600 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^5)/32, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+20*x^2+x*O(x^n));for(i=0,n,A=A+(-13*A+12+32*x+A^5)/8);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^5)/32). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (12+32*x)^(4*n+1)/13^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(-3/2 + 5*n) * (-12 + 4*(13/5)^(5/4))^(1/2 - n) / (5^(1/8) * 13^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120601 G.f. satisfies: 15*A(x) = 14 + 27*x + A(x)^6, starting with [1,3,15].

Original entry on oeis.org

1, 3, 15, 210, 3510, 65562, 1310901, 27446760, 594104940, 13187589690, 298555767279, 6867021319722, 160017552201780, 3769622456958720, 89628027015591870, 2148034269252052608, 51836638064282565579
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 + 15*x^2 + 210*x^3 + 3510*x^4 + 65562*x^5 +...
A(x)^6 = 1 + 18*x + 225*x^2 + 3150*x^3 + 52650*x^4 + 983430*x^5 +...

Crossrefs

Cf. A120588 - A120600, A120602 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+15*x - (1+x)^6)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+15*x^2+x*O(x^n));for(i=0,n,A=A+(-15*A+14+27*x+A^6)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+15*x - (1+x)^6)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (14+27*x)^(5*n+1)/15^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1/2 + 3*n) * (-14 + 5*(5/2)^(6/5))^(1/2 - n) / (2^(3/5) * 5^(9/10) * 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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