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.

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

A120603 G.f. satisfies: 16*A(x) = 15 + 27*x + A(x)^7, starting with [1,3,21].

Original entry on oeis.org

1, 3, 21, 399, 9135, 233709, 6400947, 183585897, 5443737390, 165536020650, 5133935821014, 161768728483362, 5164132704296202, 166660621950110526, 5428573285691233650, 178234125351736454070, 5892439158797172244515
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 + 21*x^2 + 399*x^3 + 9135*x^4 + 233709*x^5 +...
A(x)^7 = 1 + 21*x + 336*x^2 + 6384*x^3 + 146160*x^4 + 3739344*x^5 +...

Crossrefs

Cf. A120588 - A120602, A120604 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+16*x - (1+x)^7)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+21*x^2+x*O(x^n));for(i=0,n,A=A+(-16*A+15+27*x+A^7)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+16*x - (1+x)^7)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(7*n,n)/(6*n+1) * (15+27*x)^(6*n+1)/16^(7*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 7^(-13/12 + 2*n) * 9^n * (-245 + 32*2^(2/3)*7^(5/6))^(1/2 - n) / (2^(8/3) * 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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