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 1-3 of 3 results.

A126966 Expansion of sqrt(1 - 4*x)/(1 - 2*x).

Original entry on oeis.org

1, 0, -2, -8, -26, -80, -244, -752, -2362, -7584, -24892, -83376, -284324, -984672, -3455144, -12259168, -43908026, -158531392, -576352364, -2107982128, -7750490636, -28629222112, -106190978264, -395347083808, -1476813394916, -5533435084480, -20790762971864, -78316232088032
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Comments

Hankel transform is 2^n*(-1)^binomial(n+1, 2) = A120617(n). - Paul Barry, Feb 08 2008

Links

Crossrefs

Cf. A000108, A028329, A082590, A126967.

Programs

  • GAP
    List([0..30], n-> (-1)*Sum([0..n], j-> 2^j*Binomial(2*(n-j), n-j)/(2*(n-j) -1) )); # G. C. Greubel, Jan 29 2020
  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-2*x) )); // G. C. Greubel, Jan 29 2020
  • Maple
    a := n -> -add(2^j*binomial(2*n-2*j,n-j)/(2*n-2*j-1), j=0..n):
seq(a(n),n=0..30); # Emeric Deutsch, Mar 25 2007
# second Maple program:
CatalanNumber := n -> binomial(2*n, n)/(n+1):
a := n -> 2^n*I + CatalanNumber(n)*simplify(hypergeom([1, n + 1/2], [n + 2], 2)):
seq(a(n), n=0..26); # Peter Luschny, Aug 04 2020
# third program:
A126966 := n -> 2*binomial(2*n, n) - add(2^(n-k)*binomial(2*k,k), k=0..n):
seq(A126966(n), n = 0 .. 27); # Mélika Tebni, Mar 08 2024
  • Mathematica
    CoefficientList[Series[Sqrt[1-4*x]/(1-2*x), {x,0,30}], x] (* G. C. Greubel, Jan 31 2017 *)
  • PARI
    Vec(sqrt(1-4*x)/(1-2*x) + O(x^30)) \\ G. C. Greubel, Jan 31 2017
  • Sage
    def A126966_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(1-4*x)/(1-2*x) ).list()
A126966_list(30) # G. C. Greubel, Jan 29 2020

Formula

a(n) = -Sum_{j=0..n} ( 2^j*binomial(2n-2j, n-j)/(2n-2j-1) ). - Emeric Deutsch, Mar 25 2007
D-finite with recurrence: n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011, corrected Feb 17 2020
a(n) ~ -4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
a(n) = 2^n*i + CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 2). - Peter Luschny, Aug 04 2020
a(n) = A028329(n) - A082590(n). - Mélika Tebni, Mar 08 2024

A127144 Q(2,n), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

1, 1, 0, -3, -4, 15, 48, -105, -624, 945, 9600, -10395, -175680, 135135, 3790080, -2027025, -95235840, 34459425, 2752081920, -654729075, -90328089600, 13749310575, 3328103116800, -316234143225, -136191650918400, 7905853580625, 6131573025177600, -213458046676875, -301213549769932800
Offset: 0

Views

Author

N. J. A. Sloane, Mar 24 2007

Keywords

References

  • V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

Links

Crossrefs

A126967 interleaved with A001147.
Column 2 of A127080.
Cf. A127137, A127138, A127145.

Programs

  • Maple
    Q:= proc(n, k) option remember;
      if k<2 then 1
    elif `mod`(k,2)=0 then (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
      fi; end;
seq( Q(2, n), n=0..30); # G. C. Greubel, Jan 30 2020
  • Mathematica
    Q[n_, k_]:= Q[n, k]= If[k<2, 1, If[EvenQ[k], (n-k+1)*Q[n+1, k-1] - (k-1)*Q[n + 2, k-2], ((n-k+1)*Q[n+1, k-1] - (k-1)*(n+1)*Q[n+2, k-2])/n]]; Table[Q[2, k], {k,0,30}] (* G. C. Greubel, Jan 30 2020 *)
  • Sage
    @CachedFunction
def Q(n,k):
    if (k<2): return 1
    elif (mod(k,2)==0): return (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else: return ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
[Q(2,n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

Formula

See A127080 for e.g.f.

A126089 Expansion of e.g.f.: (1-2*x)*sqrt(1-4*x).

Original entry on oeis.org

1, -4, 4, 0, -48, -960, -20160, -483840, -13305600, -415134720, -14529715200, -564583219200, -24135932620800, -1126343522304000, -56992982228582400, -3108708121559040000, -181859425111203840000, -11359219476176732160000, -754576722346025779200000
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Crossrefs

Cf. A126967, A126079.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-2*x)*Sqrt(1-4*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Jan 25 2020
  • Mathematica
    CoefficientList[Series[(1-2*x)*Sqrt[1-4*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
  • PARI
    seq(n)={Vec(serlaplace((1-2*x)*sqrt(1-4*x + O(x*x^n))))} \\ Andrew Howroyd, Jan 25 2020

Formula

a(n) ~ -2^(2*n-3/2)*n^(n-1)/exp(n). - Vaclav Kotesovec, Jun 02 2013
D-finite with recurrence: a(n) -4*n*a(n-1) +12*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jan 24 2020
Conjecture D-finite with recurrence: (-n+4)*a(n) +2*(2*n-5)*(n-3)*a(n-1)=0. - R. J. Mathar, Jan 24 2020
Showing 1-3 of 3 results.

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

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