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.

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).

A320826 Expansion of x*(1 - 4*x)^(3/2)/(3*x - 1)^2.

Original entry on oeis.org

0, 1, 0, -3, -14, -51, -168, -521, -1542, -4365, -11740, -29439, -65670, -112273, -28344, 1018689, 6961550, 34606929, 151831044, 623095683, 2453975622, 9402575805, 35339538912, 130994480547, 480676041954, 1750847208621, 6343667488692, 22899720430251, 82466180250590
Offset: 0

Views

Author

Peter Luschny, Oct 22 2018

Keywords

Links

Crossrefs

Cf. A002421, A320825, A320827.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x*(1-4*x)^(3/2)/(1-3*x)^2)); // G. C. Greubel, Oct 27 2018
  • Maple
    c := n -> (-4)^(n-1)*binomial(3/2, n-1):
h := n -> hypergeom([2, 1 - n], [7/2 - n], 3/4):
A320826 := n -> c(n)*h(n): seq(simplify(A320826(n)), n=0..28);
  • Mathematica
    CoefficientList[Series[(x (1 -  4 x)^(3/2))/(3 x - 1)^2, {x, 0, 28}], x]
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(1-4*x)^(3/2)/(1-3*x)^2)) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = c(n)*h(n) where c(n) = Catalan(n)*(3*n*(n + 1))/(2*(2*n-5)*(2*n-3)*(2*n-1)) = (-4)^(n-1)*binomial(3/2, n-1) and h(n) = hypergeom([2, 1 - n], [7/2 - n], 3/4).
A320826(n) = A320825(n) - A320827(n).
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.