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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A320827 G.f.: -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1).

Original entry on oeis.org

-1, 1, 1, 3, 11, 41, 151, 549, 1977, 7075, 25229, 89831, 319881, 1140523, 4075321, 14603243, 52501659, 189440937, 686181711, 2495243373, 9109701699, 33388293177, 122840931891, 453622854873, 1681057537359, 6250742452125, 23316503569983, 87236431248445
Offset: 0

Views

Author

Peter Luschny, Oct 23 2018

Keywords

Links

Crossrefs

Cf. A002421, A029759, A320825, A320826.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(1-4*x)*(1-2*x)/(3*x-1))); // G. C. Greubel, Oct 27 2018
  • Maple
    ogf := x -> -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1);
ser := series(ogf(x), x, 30); seq(coeff(ser, x, k), k=0..27);
# By recurrence:
a := proc(n) option remember; if n <= 4 then return [-1,1,1,3,11][n+1] fi;
((-90+66*n-12*n^2)*a(n-2)+(30-34*n+7*n^2)*a(n-1))/((n-4)*n) end:
seq(a(n), n=0..27);
  • Mathematica
    a[n_] := (-4)^n Binomial[3/2,n]((4/3)n - 2 + Hypergeometric2F1[1,-n, 5/2 - n, 3/4]); Table[a[n], {n, 0, 27}]
CoefficientList[Series[Sqrt[1-4*x]*(1-2*x)/(3*x-1), {x, 0, 40}], x] (* G. C. Greubel, Oct 27 2018 *)
  • PARI
    x='x+O('x^40); Vec(sqrt(1-4*x)*(1-2*x)/(3*x-1)) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = (-4)^n*binomial(3/2, n)*((4/3)*n - 2 + hypergeom([1, -n], [5/2 - n], 3/4)).
D-finite with recurrence: a(n) = ((-90+66*n-12*n^2)*a(n-2) + (30-34*n+7*n^2)*a(n-1))/((n-4)*n) for n >= 5.
Expansion of -1/g.f. gives A029759.
a(n) = A320825(n) - A320826(n).

A320825 Expansion of -(10*x^2 - 6*x + 1)*sqrt(1 - 4*x)/(3*x - 1)^2.

Original entry on oeis.org

-1, 2, 1, 0, -3, -10, -17, 28, 435, 2710, 13489, 60392, 254211, 1028250, 4046977, 15621932, 59463209, 224047866, 838012755, 3118339056, 11563677321, 42790868982, 158180470803, 584617335420, 2161733579313, 8001589660746, 29660171058675, 110136151678696, 409773163539325
Offset: 0

Views

Author

Peter Luschny, Oct 22 2018

Keywords

Links

Crossrefs

Cf. A126079, A320826, A320827.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(-(10*x^2-6*x+1)*Sqrt(1-4*x)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018
  • Maple
    c := n -> catalan(n)*(n-3)*(n+1)/((2*n-3)*(2*n-1)):
h := n -> hypergeom([1, -n], [5/2 - n], 3/4): A320825 := n -> c(n)*h(n):
seq(simplify(A320825(n)), n=0..28);
  • Mathematica
    CoefficientList[Series[-(10x^2 - 6x + 1) Sqrt[1 - 4x]/(3x - 1)^2, {x, 0, 28}], x]
  • PARI
    x='x+O('x^40); Vec(-(10*x^2-6*x+1)*sqrt(1-4*x)/(1-3*x)^2) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = c(n)*h(n) where c(n) = Catalan(n)*((n-3)*(n+1))/((2*n-3)*(2*n-1)) and h(n) = hypergeom([1, -n], [5/2 - n], 3/4).
A320825(n) = A320826(n) + A320827(n).
D-finite with recurrence: n*a(n) +(-13*n+15)*a(n-1) +4*(16*n-37)*a(n-2) +2*(-71*n+245)*a(n-3) +60*(2*n-9)*a(n-4)=0. - R. J. Mathar, Jan 23 2020
Showing 1-2 of 2 results.

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