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.

A108623 G.f. satisfies x = A(x)*(1-A(x))/(1-A(x)-(A(x))^2).

Original entry on oeis.org

1, 0, -1, -1, 1, 4, 3, -8, -23, -10, 67, 153, 9, -586, -1081, 439, 5249, 7734, -7941, -47501, -53791, 105314, 430119, 343044, -1249799, -3866556, -1730017, 13996097, 34243897, 1947204, -150962373, -296101864, 121857185, 1582561870
Offset: 1

Views

Author

Christian G. Bower, Jun 12 2005

Keywords

Comments

Row sums of inverse of Riordan array (1/(1-x-x^2), x*(1-x)/(1-x-x^2)) (Cf. A053538). - Paul Barry, Nov 01 2006

Examples

			G.f. = x - x^3 - x^4 + x^5 + 4*x^6 + 3*x^7 - 8*x^8 - 23*x^9 - 10*x^10 + ...

Links

Crossrefs

Except for signs, same as A108624.
Cf. A039978, A105523, A212804.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 41);
Coefficients(R!( (1+x-Sqrt(1-2*x+5*x^2))/(2*(1-x)) )); // G. C. Greubel, Oct 20 2023
  • Maple
    # Using function CompInv from A357588.
CompInv(34, n -> ifelse(n=-1, 1, combinat:-fibonacci(n-2))); # Peter Luschny, Oct 05 2022
  • Mathematica
    CoefficientList[Series[(1+x-Sqrt[1-2*x+5*x^2])/(2*x*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 2 x + 5 x^2]) / (2 (1 - x)), {x, 0, n}]; (* Michael Somos, May 19 2014 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ (x - x^2) / (1 - x - x^2), {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, May 19 2014 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x + 5*x^2 + x^2 * O(x^n))) / (2 * (1 - x)), n))}; /* Michael Somos, May 19 2014 */
  • PARI
    {b(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 - x - x^2) + x * O(x^n)), n))}; /* Michael Somos, May 19 2014 */
  • SageMath
    def A108623_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)) ).list()
a=A108623_list(41); a[1:] # G. C. Greubel, Oct 20 2023

Formula

Binomial transform of A105523. - Paul Barry, Nov 01 2006
G.f.: (1+x-sqrt(1-2*x+5*x^2))/(2*(1-x)). - Paul Barry, Nov 01 2006
Conjecture: n*a(n) +3*(1-n)*a(n-1) +(7*n-18)*a(n-2) +5*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 15 2011
Lim sup_{n->infinity} |a(n)|^(1/n) = sqrt(5). - Vaclav Kotesovec, Feb 08 2014
Series reversion of g.f. of A212804. - Michael Somos, May 19 2014
G.f.: x / (1 - x + x /(1 - x / (1 - x + x / (1 - x / ...)))). - Michael Somos, May 19 2014
0 = a(n)*(25*a(n+1) - 50*a(n+2) + 45*a(n+3) - 20*a(n+4)) + a(n+1)*(-20*a(n+1) + 34*a(n+2) - 44*a(n+3) + 25*a(n+4)) + a(n+2)*(12*a(n+2) - 2*a(n+3) - 6*a(n+4)) + a(n+3)*(a(n+4)) if n>=0. - Michael Somos, May 19 2014

A039980 An example of a d-perfect sequence.

Original entry on oeis.org

1, 0, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 0, 1, 2, 2, 2, 0, 0, 2, 2, 1, 0, 0, 1, 0, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 0, 1, 1, 2, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 1, 1, 1, 2, 0, 2, 2, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 2, 0, 2, 2, 1, 0, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 0, 1, 1, 2, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

  • D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

Crossrefs

Cf. A108624.

Formula

a(n) = A108624(n) mod 3. - Christian G. Bower, Jun 12 2005

Extensions

More terms from Christian G. Bower, Jun 12 2005

A202327 Triangle read by rows, T(n, k) is the coefficient of x^n in expansion of ((-1 - x + sqrt(1 + 2*x + 5*x^2)) /2)^k.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 2, 1, -3, 1, -3, 4, 3, -4, 1, -1, -10, 5, 6, -5, 1, 11, 4, -21, 4, 10, -6, 1, -15, 28, 21, -35, 0, 15, -7, 1, -13, -64, 42, 56, -50, -8, 21, -8, 1, 77, 9, -162, 36, 114, -63, -21, 28, -9, 1, -86, 230, 135, -312, -15, 198, -70, -40, 36, -10, 1
Offset: 1

Views

Author

Vladimir Kruchinin, Dec 17 2011

Keywords

Examples

			   1;
  -1,   1;
   0,  -2,   1;
   2,   1,  -3,   1;
  -3,   4,   3,  -4,   1;
  -1, -10,   5,   6,  -5,   1;
  11,   4, -21,   4,  10,  -6,   1;

Crossrefs

Cf. A007440 (1st column), A108624 (row sums).

Programs

  • Maple
    # Assuming offset = 0.
T := (n,k) -> (-1)^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], -4): for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, May 19 2021
  • Maxima
    T(n,k):=(k*sum(binomial(j,-n-k+2*j)*(-1)^(j-k)*binomial(n,j),j,0,n))/n;

Formula

T(n, k) = (k/n) * Sum_{j=0..n} (-1)^(j-k) * binomial(n,j) * binomial(j,-n-k+2*j).
T(n, k) = binomial(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [k + 2], -4)*(-1)^(n - k), assuming offset = 0. - Peter Luschny, May 19 2021

Extensions

More terms from Sean A. Irvine, Mar 03 2021
Showing 1-3 of 3 results.

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

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