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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A374509 Expansion of 1/(1 - 2*x + 5*x^2)^(7/2).

Original entry on oeis.org

1, 7, 14, -42, -294, -462, 1386, 7722, 9009, -37037, -160160, -123760, 835380, 2848860, 1046520, -16550520, -45140865, 3533145, 296447690, 648593330, -393463070, -4895709390, -8489647530, 10975099590, 75528298755, 100311659721, -230350834728, -1097798696456
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2024

Keywords

Crossrefs

Cf. A098331, A102840, A374508.
Cf. A374506.

Programs

  • Mathematica
    a[n_]:= Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, -4]/6!; Array[a,28,0] (* Stefano Spezia, Jul 10 2024 *)
  • PARI
    a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, (-1)^k*binomial(n+3, n-2*k)*binomial(2*k+3, k));

Formula

a(0) = 1, a(1) = 7; a(n) = ((2*n+5)*a(n-1) - 5*(n+5)*a(n-2))/n.
a(n) = (binomial(n+6,3)/20) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = Pochhammer(n+1, 6)*hypergeom([(1-n)/2, -n/2], [4], -4)/6!. - Stefano Spezia, Jul 10 2024
a(n) = (-1)^n * Sum_{k=0..n} 2^k * (5/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 23 2025

A098338 Expansion of 1/sqrt(1-6x+13x^2).

Original entry on oeis.org

1, 3, 7, 9, -21, -207, -911, -2769, -5213, 2457, 74997, 400491, 1409109, 3323583, 2219343, -27453951, -186624333, -750905127, -2088947819, -2955863589, 8506703569, 86421384387, 401183114163, 1280139325101, 2522745571021
Offset: 0

Views

Author

Paul Barry, Sep 03 2004

Keywords

Comments

Binomial transform of A098335. Second binomial transform of A098331.
Central coefficients of (1+3x-x^2)^n.

Links

Programs

  • Maple
    f:= gfun:-rectoproc({(13*n+13)*a(n)+(-9-6*n)*a(n+1)+(n+2)*a(n+2), a(0)=1, a(1)=3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2017
  • Mathematica
    CoefficientList[Series[1/Sqrt[1-6*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)

Formula

E.g.f.: exp(3*x)*BesselI(0, 2*I*x), I=sqrt(-1).
a(n) = Sum{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-9)^(-k).
a(n) = Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*3^n*(-9)^(-k).
D-finite with recurrence: n*a(n) +3*(1-2*n)*a(n-1) +13*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
Recurrence follows from the differential equation (13x-3) g(x) + (13x^2-6x+1) g'(x) = 0 satisfied by the generating function. - Robert Israel, Mar 02 2017
Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Sep 29 2013

A112874 Numbers n such that the coefficient of x^n in (x^2-x-1)^n is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 26, 160, 3787, 31877
Offset: 1

Views

Author

T. D. Noe, Sep 29 2005

Keywords

Comments

n=31877 yields a probable prime; the others have been proved prime. These n were also found by Eric Weisstein, who found no other n<100000.

Links

Crossrefs

Cf. A098331 (coefficient of x^n in (x^2-x-1)^n).

Programs

  • Mathematica
    Select[Range[200], PrimeQ[Coefficient[Expand[(x^2-x-1)^# ], x, # ]]&]
Previous Showing 11-13 of 13 results.

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

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