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 1-2 of 2 results.

A208034 G.f.: exp( Sum_{n>=1} 2*Pell(n)^2 * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 6, 26, 122, 602, 3062, 15906, 83906, 447842, 2412566, 13094490, 71513210, 392592410, 2164815590, 11982792386, 66548673282, 370672213826, 2069974290726, 11586244722202, 64986102400122, 365183031749722, 2055594717395926, 11588727763937506, 65425688924696002
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjectures: For all positive integers k,
(1) exp( Sum_{n>=1} 2*Pell(n)^(2*k) * x^n/n ) is an integer series;
(2) exp( Sum_{n>=1} 2*Pell(n)^(2*k-1) * x^n/n ) is NOT an integer series;
(3) exp( Sum_{n>=1} Pell(n)^(2*k) * x^n/n ) is NOT an integer series.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 122*x^4 + 602*x^5 + 3062*x^6 + ...
such that, by definition,
log(A(x))/2 = x + 2^2*x^2/2 + 5^2*x^3/3 + 12^2*x^4/4 + 29^2*x^5/5 + 70^2*x^6/6 + 169^2*x^7/7 + 408^2*x^8/8 + ... + Pell(n)^2*x^n/n + ...

Links

Crossrefs

Cf. A000129, A208055, A208056, A204061, A204062.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)
  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^2*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    seq(n)={Vec(sqrt(1 + x + O(x^n)) / sqrt(sqrt(1 - 6*x + x^2 + O(x^n))))} \\ Andrew Howroyd, Feb 25 2018

Formula

G.f.: sqrt(1 + x) / (1 - 6*x + x^2)^(1/4).
a(n) ~ 2^(1/8) * (1 + sqrt(2))^(2*n) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 31 2024

A208056 G.f.: exp( Sum_{n>=1} 2*Pell(n)^(2*n) * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 10450, 215011842, 168283323489554, 4613762736903044410402, 4429409381416783893511092430530, 147401742703370819998531165821635082467298, 169293247178836261713452084817353169649400098579929282
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/2) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 10450*x^3 + 215011842*x^4 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^6*x^3/3 + 12^8*x^4/4 + 29^10*x^5/5 + 70^12*x^6/6 + 169^14*x^7/7 +...+ Pell(n)^(2*n)*x^n/n +...

Crossrefs

Cf. A000129, A211891, A208034, A208055, A204061, A204062.

Programs

  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^(2*m)*x^m/m) +x*O(x^n)),n)}
for(n=0,15,print1(a(n),", "))
Showing 1-2 of 2 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.