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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A104497 Expansion of sqrt(1-8x)/sqrt(1-4x).

Original entry on oeis.org

1, -2, -10, -52, -282, -1596, -9412, -57640, -364922, -2376812, -15852204, -107821656, -745342500, -5221954776, -36997822536, -264620356944, -1907962439994, -13852652486220, -101186612941084, -743057485099384, -5482375128919820, -40620301416309128
Offset: 0

Views

Author

Paul Barry, Mar 11 2005, Mar 07 2008

Keywords

Comments

Convolution of A098579 and A000984.
Hankel transform is (-2)^n*A002315(n).

Crossrefs

Cf. A000108, A104498.

Formula

Moment representation: a(n) = (1/Pi)*Integral_{x = 4..8} (x^n*sqrt((8 - x)*(x - 4))/(x*(4 - x))) dx + sqrt(2)*0^n;
Also a(n) = C(2*n, n) - Sum_{k = 1..n} 2^(k+1)*C(2*k-2, k-1)*C(2*n-2*k, n-k)/k.
From Peter Bala, Feb 03 2024: (Start)
a(n) = - 2*A104498(n) for n >= 1.
G.f.: c(-x/(1 - 8*x)) / c(x/(1 - 4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
P-recursive: n*a(n) = (12*n - 14)*a(n-1) - 32*(n - 2)*a(n-2) with a(0) = 1 and a(1) = -2. (End)

A261196 Expansion of sqrt(8*x + sqrt(1 + 64*x^2)).

Original entry on oeis.org

1, 4, 8, -32, -160, 896, 5376, -33792, -219648, 1464320, 9957376, -68796416, -481574912, 3408068608, 24343347200, -175272099840, -1270722723840, 9268801044480, 67971207659520, -500840477491200, -3706219533434880, 27531916534087680, 205237923254108160
Offset: 0

Views

Author

Werner Schulte, Aug 11 2015

Keywords

Comments

Signs are important to distinguish this from (for example) A098579.
Sqrt(A(x)) = 1 + 2*x + 2*x^2 - 20*x^3 - 42*x^4 + 572*x^5 ... defines another (new) integer sequence.

Examples

			A(x) = 1 + 4*x + 8*x^2 - 32*x^3 - 160*x^4 + 896*x^5 + 5376*x^6 ...
A(x)^2 = 1 + 8*x + 32*x^2 + 0*x^3 - 512*x^4 + 0*x^5 +16384*x^6 ...
A(x)^4 = 1 + 16*x + 128*x^2 + 512*x^3 + 0*x^4 -8192*x^5 + 0*x^6 ...

Links

Crossrefs

Cf. A000108, A002420, A098579, A182122.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(8*x+Sqrt(1+64*x^2)))); // G. C. Greubel, Aug 12 2018
  • Mathematica
    CoefficientList[Series[Sqrt[8 x + Sqrt[1 + 64 x^2]], {x, 0, 45}], x] (* Vincenzo Librandi, Aug 12 2015 *)
  • PARI
    Vec(sqrt(8*x + sqrt(1 + 64*x^2))) \\ M. F. Hasler, Aug 14 2015

Formula

G.f. A(x) satisfies: A(x)^4 = 1 + 16*x*A(x)^2.
A(x) = sqrt(G(4*x)), where G(x) is the g.f. of A182122.
A(x) * A(-x) = 1.
A(x) = sqrt(1 + 8*x + 32*x^2*C(-16*x^2)), where C(x) is the g.f. of A000108.
a(n) = A002420(n)*2^n*(-1)^(n*(n+1)/2). - M. F. Hasler, Aug 14 2015
Conjecture D-finite with recurrence: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) +16*(2*n-3)*(2*n-5)*a(n-2) +16*(2*n-5)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jun 07 2016
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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