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.

A049171 Revert transform of 2*x*(1-x)-x/(1+x).

Original entry on oeis.org

1, 1, 3, 9, 33, 125, 503, 2081, 8849, 38345, 168875, 753401, 3398177, 15469493, 70984559, 327982529, 1524644897, 7125440913, 33459931155, 157794990633, 747021246817, 3548843286829, 16912921740775, 80836929471329, 387397148131889, 1861088017162457
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Sign diagram of generating sequence: +++-------------...
The sequences A049171 to A049189 are defined by series reversion of a sequence with rational (ordinary) generating function g(x). Solving g(x)=y for x yields algebraic equations for x, so the sequences have P-finite recurrences. - R. J. Mathar, Jul 24 2023

Links

Programs

  • Mathematica
    Table[Sum[Binomial[3*k,k]*Binomial[n-1+k,3*k]/(2k+1)*2^k,{k,0,Floor[(n-1)/2]}],{n,1,20}] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n+1) = Sum_{k=0..floor(n/2)} A108759(n,k)*2^k. - Philippe Deléham, Dec 08 2009
Recurrence: 4*(n-1)*n*a(n) = 2*(n-1)*(5*n-6)*a(n-1) + 3*(16*n^2 - 67*n + 69)*a(n-2) + (25*n^2 - 169*n + 285)*a(n-3) + (n-4)*(2*n-9)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(sqrt(3)-1)*((5+3*sqrt(3))/2)^n/(2*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Extensions

NAME corrected by R. J. Mathar, Jul 23 2023

A049186 Revert transform of 2*x*(1-x-x^3-x^5+x^6)-x/(1+x).

Original entry on oeis.org

1, 1, 3, 11, 45, 197, 901, 4259, 20643, 102035, 512365, 2606421, 13403715, 69568071, 363946211, 1917148173, 10160087333, 54132552109, 289790960735, 1557980372127, 8408281616305, 45536927325809, 247397689058521
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Sign diagram of generating sequence: +++-+-++--------...
2*x*(1-x-x^3-x^5+x^6)-x/(1+x) = x -x^2 -x^3 -x^4 -x^5 -x^6 +x^7 +x^8 -x^9 +x^10-+... = y <-> x= y +y^2 +3*y^3 +11*y^4 +45*y^5+... - R. J. Mathar, Jul 21 2023

Links

Crossrefs

Cf. A049189.

Programs

  • Maple
    Order := 80:
g := 2*x*(1-x-x^3-x^5+x^6)-x/(1+x) ;
solve(series(g, x)=y, x) :
convert(%, polynom) :
seq(coeff(%, y, i), i=0..Order-1) ; # R. J. Mathar, Jul 24 2023
  • PARI
    my(x='x+O('x^40)); Vec(serreverse(2*x*(1-x-x^3-x^5+x^6)-x/(1+x))) \\ Michel Marcus, Jul 21 2023

Extensions

NAME corrected by R. J. Mathar, Jul 21 2023
Showing 1-2 of 2 results.

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

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