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.

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A126984 Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -2, 2, -4, 2, -12, -12, -72, -190, -700, -2308, -8120, -28364, -100856, -360792, -1301904, -4727358, -17268636, -63405012, -233885784, -866327748, -3220976616, -12016209192, -44966763504, -168750724428, -634935132312, -2394717424552, -9051945482032
Offset: 0

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Author

Philippe Deléham, Mar 21 2007

Keywords

Comments

Hankel transform is (-2)^n.
Hankel transform omitting first term is (-2)^n omitting first term. Hankel transform omitting first two terms is 2*(-1)^n*A014480(n). - Michael Somos, May 16 2022

Links

Crossrefs

Cf. A000108, A014480, A039599, A268600, A268601.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2-Sqrt(1-4*x)) )); // G. C. Greubel, May 28 2019
  • Maple
    c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+2*x*c),x=0,32): seq(coeff(ser,x,n),n=0..30); # Emeric Deutsch, Mar 24 2007
  • Mathematica
    CoefficientList[Series[1/(2-Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 28 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/(2-sqrt(1-4*x))) \\ G. C. Greubel, May 28 2019
  • Sage
    (1/(2-sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

Formula

a(n) = Sum_{k=0..n} A039599(n,k)*(-3)^k.
G.f.: 1/(2 - sqrt(1-4*x)). - G. C. Greubel, May 28 2019
(-1)^n*a(n) = A268600(n) - A268601(n). - Michael Somos, May 16 2022
D-finite with recurrence 3*n*a(n) +2*(-4*n+9)*a(n-1) +8*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k = 0..n} A009766(n-1, k)*(-2)^(n-k) for n >= 1. - Peter Bala, Jun 18 2025

Extensions

Corrected and extended by Emeric Deutsch, Mar 24 2007

A268601 Expansion of 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).

Original entry on oeis.org

0, 0, 2, 8, 34, 120, 468, 1680, 6530, 23960, 93532, 348656, 1366260, 5149872, 20238696, 76907808, 302903874, 1158168792, 4569270156, 17555689008, 69356428284, 267518448912, 1058057586456, 4094231982048, 16208177203764, 62887835652720, 249156625186328, 968943740083040, 3841488520364200, 14968574892499040, 59379627044952528
Offset: 0

Views

Author

Ran Pan, Feb 08 2016

Keywords

Comments

a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is odd. Details can be found in Section 4.1 in Pan and Remmel's link.

Links

Crossrefs

Cf. A268462, A268586, A268587, A268598, A268599.

Programs

  • PARI
    x = 'x + O('x^30); concat(vector(2), Vec(1/(2*sqrt(1-4*x)) - 1/(4 - 2*sqrt(1+4*x)))) \\ Michel Marcus, Feb 11 2016

Formula

a(n) = binomial(2*n,n) - A268600(n).
G.f.: 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
Conjecture D-finite with recurrence: 3*n*(n-1)*a(n) -8*(n-1)*(5*n-12)*a(n-1) +4*(28*n-73)*a(n-2) +160*(2*n-5)*(2*n-7)*a(n-3) -192*(2*n-5)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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