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.

A120598 G.f. satisfies: 30*A(x) = 29 + 125*x + A(x)^5, starting with [1,5,10].

Original entry on oeis.org

1, 5, 10, 90, 825, 8445, 92820, 1066740, 12670635, 154308775, 1916370170, 24177471370, 309007779015, 3992428316835, 52059968802000, 684240882022800, 9055282215370050, 120563388411386850, 1613785688724362400
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 + 5*x + 10*x^2 + 90*x^3 + 825*x^4 + 8445*x^5 +...
A(x)^5 = 1 + 25*x + 300*x^2 + 2700*x^3 + 24750*x^4 + 253350*x^5 +...

Crossrefs

Cf. A120588 - A120597, A120599 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+30*x - (1+x)^5)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+5*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-30*A+29+125*x+A^5)/25);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+30*x - (1+x)^5)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (29+125*x)^(4*n+1)/30^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 5^(-1/2 + 3*n) * (-29 + 24*6^(1/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120600 G.f. satisfies: 7*A(x) = 6 + x + A(x)^6, starting with [1,1,15].

Original entry on oeis.org

1, 1, 15, 470, 18390, 805806, 37828981, 1860433080, 94614523740, 4935081398830, 262560448214031, 14193030016877406, 777315341935068820, 43039297954660894560, 2405249540028525971070, 135492504636185052358656
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 + 15*x^2 + 470*x^3 + 18390*x^4 + 805806*x^5 +...
A(x)^6 = 1 + 6*x + 105*x^2 + 3290*x^3 + 128730*x^4 + 5640642*x^5 +...

Crossrefs

Cf. A120588 - A120599, A120601 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[1+7*x - (1+x)^6, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+x+15*x^2+x*O(x^n));for(i=0,n,A=A-7*A+6+x+A^6);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion(1+7*x - (1+x)^6). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (6+x)^(5*n+1)/7^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-6 + 5*(7/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 7^(2/5) * n^(3/2) * sqrt(5*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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