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 1-5 of 5 results.

A098332 Expansion of 1/sqrt(1 - 2*x + 9*x^2).

Original entry on oeis.org

1, 1, -3, -11, 1, 81, 141, -363, -1791, -479, 13597, 29877, -54911, -353807, -223443, 2539989, 6806529, -8302527, -73999299, -73313931, 489731841, 1584548241, -1110170163, -15812965611, -21391839999, 94696016481
Offset: 0

Views

Author

Paul Barry, Sep 03 2004

Keywords

Comments

Central coefficients of (1 + x - 2*x^2)^n.
Binomial transform of 1/sqrt(1+8*x^2), or (1,0,-4,0,24,0,...).
Binomial transform is A098336.

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, ex. 7.56, p. 575.

Links

Crossrefs

Cf. A126869, A012000, A116091, A098341.

Programs

  • Maple
    a := n -> hypergeom([1/2-n/2, -n/2], [1], -8);
seq(round(evalf(a(n),99)),n=0..30); # Peter Luschny, Sep 18 2014
  • Mathematica
    Table[(-3)^n*LegendreP[n,-1/3],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1 - 2*x + 9*x^2], {x,0,50}], x] (* G. C. Greubel, Feb 18 2017 *)
  • PARI
    x='x+O('x^25); Vec(1/sqrt(1 - 2*x + 9*x^2)) \\ G. C. Greubel, Feb 18 2017

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-2)*x);
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * (-2)^k.
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-k,k) * (-2)^k.
a(n) = (-1)^n * Sum_{k=0..n} binomial(n,k)^2 * (-2)^k.
G.f.: A(x) = 1/(2*T(0)+3*x-1) where T(k) = 1 - 2*x/(1 + x/T(k+1)); (continued fraction, 2-step ). - Sergei N. Gladkovskii, Aug 23 2012
D-finite with recurrence: a(n+2) = ((2*n+3)*a(n+1))/(n+2) - (9*(n+1)*a(n))/(n+2) with a(0)=1, a(1)=1. (See Graham, Knuth, and Patashnik). - Alexander R. Povolotsky, Aug 23 2012
a(n) = hypergeom([1/2-n/2, -n/2], [1], -8). - Peter Luschny, Sep 18 2014
a(n) = (3/2)*(9/2)^n*Sum_{k >= 0} (-1/2)^k*binomial(n+k,k)^2. - Peter Bala, Mar 02 2017

A116091 Expansion of 1/sqrt(1+4*x+16*x^2).

Original entry on oeis.org

1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228, -197372, -267896, 3921724, -11126936, -15347432, 225505648, -643622906, -897078476, 13214495764, -37869162392, -53170602284, 784672445368, -2255295815192, -3183829452272, 47051201187676, -135537088268792, -192142210448216
Offset: 0

Views

Author

Paul Barry, Feb 04 2006

Keywords

Comments

Fourth binomial transform is expansion of 1/sqrt(1-4*x+16*x^2), A012000.

Links

Crossrefs

Cf. A012000, A098332, A126869.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+16*x^2) )); // G. C. Greubel, May 09 2019
  • Maple
    a := n -> (-4)^n*hypergeom([-n,1+n],[1],1/4);
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 09 2016
  • Mathematica
    Table[4^n*LegendreP[n,-1/2],{n,0,30}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1+4x+16x^2],{x,0,30}],x] (* Harvey P. Dale, Jun 08 2015 *)
  • PARI
    Vec(1/sqrt(1+4*x+16*x^2+O(x^30))) \\ M. F. Hasler, Aug 25 2012
  • Sage
    (1/sqrt(1+4*x+16*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

Formula

E.g.f.: exp(-2*x)*Bessel_I(0, 2*sqrt(-3)*x).
a(n) = Sum_{k=0..n} C(n,k)^2*(-3)^k.
O.g.f.: P(-1/2,4*x) with the o.g.f. P(x,z):=1/sqrt(1-2*x*z+z^2) for the Legendre polynomials. Wolfdieter Lang, Mar 10 2011.
G.f. A(x) = 1/(2*T(0)-4*x-1) where T(k)= 1 + 3*x/(1 - x/T(k+1)); (continued fraction, 2-step ). - Sergei N. Gladkovskii, Aug 23 2012
D-finite with recurrence: a(n+2) = -(16*(n+1)*a(n))/(n+2) - (2*(2*n+3)*a(n+1))/(n+2) with a(0)=1, a(1)=-2. - Alexander R. Povolotsky, Aug 23 2012
a(n) = (-4)^n*hypergeom([-n, 1+n], [1], 1/4). - Peter Luschny, May 09 2016
From Peter Bala, Nov 30 2021: (Start)
a(n) = (-4)^n^P(n,1/2), where P(n,x) is the n-th Legendre polynomial.
a(n) = (4/3)*(16^n)*Sum_{k >= n} C(k,n)^2*(-1/3)^k.
a(n) = (-3)^n*hypergeom([-n, -n], [1], -1/3).
a(n) = (4/3)*(-16/3)^n*hypergeom([n+1, n+1], [1], -1/3).
a(n) = [x^n] ((1 + x)*(3 - x))^n. (End)

A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).

Original entry on oeis.org

1, 1, -5, -17, 19, 211, 181, -2015, -5837, 12259, 91585, 29965, -1033955, -2347955, 7953115, 43864543, -11941037, -559875245, -942036911, 5060812717, 21502740649, -20676139991, -307241918945, -344022187613
Offset: 0

Views

Author

Paul Barry, Sep 03 2004

Keywords

Comments

Central coefficients of (1 + x - 3x^2)^n.
Binomial transform of 1/sqrt(1+12x^2), or (1,0,-6,0,54,0,-540,...).
Binomial transform is A012000.

Links

Crossrefs

Cf. A098331, A098332, A098334.

Programs

  • Maple
    a := n -> hypergeom([1/2 - n/2, -n/2], [1], -12):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Mar 19 2018
  • Mathematica
    CoefficientList[Series[1/Sqrt[1-2*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
  • PARI
    x='x+O('x^99); Vec(1/(1-2*x+13*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-3)x);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)(-3)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)(-3)^k.
D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + 13*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Feb 09 2014
a(n) = (sqrt(13))^n*P(n,1/sqrt(13)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018
a(n) = hypergeom([1/2 - n/2, -n/2], [1], -12). - Peter Luschny, Mar 19 2018

A098341 Expansion of 1/sqrt(1 - 6*x + 25*x^2).

Original entry on oeis.org

1, 3, 1, -45, -255, -477, 2689, 25203, 82945, -90045, -2379519, -11581677, -12063999, 197669475, 1423716225, 3645266355, -12180238335, -156702949245, -626511576575, 51239061075, 15179398450945, 87687927568035, 151934475887745
Offset: 0

Views

Author

Paul Barry, Sep 03 2004

Keywords

Comments

Central coefficients of (1 + 3*x - 4*x^2)^n.
(-1)^n*a(n) is the sum of squares of coefficients of (1+2*i*x)^n where i=sqrt(-1) (see PARI code). - Joerg Arndt, Jul 06 2011
Binomial transform of A098337.
Second binomial transform of A098334.

Links

Crossrefs

Cf. A098332, A012000, A116091, A126869.

Programs

  • Mathematica
    Table[(-5)^n*LegendreP[n,-3/5],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1-6x+25x^2],{x,0,30}],x] (* Harvey P. Dale, Aug 22 2014 *)
  • PARI
    a(n)={local(v=Vec((1+2*I*x)^n)); (-1)^n*sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */
  • PARI
    a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2-imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */
  • Sage
    A098341 = lambda n: (-1)^n*hypergeometric([-n,-n], [1], -4)
[Integer(A098341(n).n(100)) for n in (0..22)] # Peter Luschny, Sep 23 2014

Formula

E.g.f.: exp(3*x)*BesselI(0, 4*I*x), I=sqrt(-1).
a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)^2*(-4)^k.
a(n) = (-1)^n*hypergeometric([-n,-n], [1], -4). - Peter Luschny, Sep 23 2014
D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 27 2014
From Peter Bala, Nov 28 2021: (Start)
a(n) = (5^n)*P(n,3/5), where P(n,x) is the n-th Legendre polynomial.
a(n) = [x^n] ((1 - x)*(1 + 4*x))^n.
a(n) = 5^(2*n+1)*Sum_{k >= n} (-1)^(n+k)*binomial(k,n)^2*(1/4)^(k+1).
a(n) = (5/4)*(25/4)^n*hypergeom([n+1, n+1], [1], -1/4). (End)

A098340 Expansion of 1/sqrt(1 - 6x + 21x^2).

Original entry on oeis.org

1, 3, 3, -27, -189, -567, 189, 11259, 59859, 129033, -395847, -4730481, -19580211, -21264201, 258785523, 1917734373, 6051991059, -2659507911, -135544952151, -738957668337, -1618780564359, 5297724346923, 63513121347063, 266379249285873, 286776522444861, -3683959713627417
Offset: 0

Views

Author

Paul Barry, Sep 03 2004

Keywords

Comments

Binomial transform of A012000. Second binomial transform of A098333.
Central coefficients of (1 + 3x - 3x^2)^n.

Links

Crossrefs

Cf. A012000, A098333.

Programs

  • Maple
    a := n -> 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Mar 19 2018
  • Mathematica
    CoefficientList[Series[1/Sqrt[1-6x+21x^2],{x,0,30}],x] (* Harvey P. Dale, Oct 07 2012 *)
  • PARI
    my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x+21*x^2)) \\ Jinyuan Wang, Sep 08 2019

Formula

E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(-3)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-3)^(-k).
a(n) = 3^n*Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-3)^(-k).
D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 21*(n-1)*a(n-2) = 0. - R. J. Mathar, Sep 26 2012
Lim sup n->infinity |a(n)|^(1/n) = sqrt(21). - Vaclav Kotesovec, Sep 29 2013
a(n) = 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3). - Peter Luschny, Mar 19 2018
Showing 1-5 of 5 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.