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

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

0,2

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

Example

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 + ...

Mathematica

CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)

Prog

(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

Crossrefs

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

Sequence in context: A358107 A288606 A377501 * A092880 A370185 A192808

Adjacent sequences: A208031 A208032 A208033 * A208035 A208036 A208037

Keyword

nonn

Author

Paul D. Hanna, Feb 22 2012

Status

approved