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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A210628 Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.

Original entry on oeis.org

1, -1, 0, -1, 0, -2, 0, -5, 0, -14, 0, -42, 0, -132, 0, -429, 0, -1430, 0, -4862, 0, -16796, 0, -58786, 0, -208012, 0, -742900, 0, -2674440, 0, -9694845, 0, -35357670, 0, -129644790, 0, -477638700, 0, -1767263190, 0, -6564120420, 0, -24466267020, 0
Offset: 0

Views

Author

Michael Somos, Mar 25 2012

Keywords

Comments

Except for the leading term, the sequence is equal to -A097331(n). - Fung Lam, Mar 22 2014

Examples

			G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ...

Links

Crossrefs

Cf. A000108, A001405, A090192, A097331, A126930, A210736.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // G. C. Greubel, Aug 11 2018
  • Mathematica
    CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* Bruno Berselli, Mar 25 2012 *)
a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];
  • Maxima
    makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); /* Bruno Berselli, Mar 25 2012 */
  • PARI
    {a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};
  • PARI
    {a(n) = if( n<1, n==0, polcoeff( serreverse( -x / (1 + x^2) + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};

Formula

G.f.: 1 - (2*x) / (1 + sqrt( 1 - 4*x^2)) = 1 - (1 - sqrt( 1 - 4*x^2)) / (2*x).
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).
G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.
G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).
G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).
G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).
G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).
G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - Michael Somos, Jan 02 2013
a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).
Convolution inverse of A210736. - Michael Somos, Jan 02 2013
G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 17 2014
For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - Fung Lam, Mar 17 2014

A201093 Triangle T(n,k), read by rows, given by (0,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,...) DELTA (1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 3, 4, 1, 0, 2, 2, 4, 6, 5, 1, 0, 0, 4, 6, 8, 10, 6, 1, 0, 5, 5, 9, 13, 15, 15, 7, 1, 0, 0, 10, 15, 20, 25, 26, 21, 8, 1, 0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Nov 26 2011

Keywords

Comments

Riordan array (1,xf(x)) where f(x) is g.f. of A097331.
Row sums are in A001405.

Examples

			Triangle begins :
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 0, 2, 3, 4, 1
0, 2, 2, 4, 6, 5, 1
0, 0, 4, 6, 8, 10, 6, 1
0, 5, 5, 9, 13, 15, 15, 7, 1
0, 0, 10, 15, 20, 25, 26, 21, 8, 1
0, 14, 14, 24, 34, 41, 45, 42, 28, 9, 1
0, 0, 28, 42, 56, 70, 78, 77, 64, 36, 10, 1
0, 42, 42, 70, 98, 120, 136, 140, 126, 93, 45, 11, 1

Crossrefs

Cf. Diagonals : A000012, A001477, A000217, A000215,
Columns :A000007, A097331, A089408

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001405(n), A098617(n) for x = 0,1,2 respectively.
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