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
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 + ...
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CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)
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{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),", "))
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seq(n)={Vec(sqrt(1 + x + O(x^n)) / sqrt(sqrt(1 - 6*x + x^2 + O(x^n))))} \\ Andrew Howroyd, Feb 25 2018
A208055
G.f.: exp( Sum_{n>=1} 2*Pell(n)^4 * x^n/n ), where Pell(n) = A000129(n).
Original entry on oeis.org
1, 2, 18, 450, 11362, 311426, 8857426, 259072706, 7730804098, 234255654466, 7184570715602, 222512186923010, 6947171244623714, 218374183252085826, 6903938704875627410, 219355658720815861378, 6999679608428089841154, 224210965624588803552642
Offset: 0
G.f.: A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + 11362*x^4 + 311426*x^5 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^4*x^3/3 + 12^4*x^4/4 + 29^4*x^5/5 + 70^4*x^6/6 + 169^4*x^7/7 + 408^4*x^8/8 +...+ Pell(n)^4*x^n/n +...
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{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)^4*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
A211891
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, 14, 682, 236826, 525175434, 7101054148862, 575978478770467714, 277997363115795461721154, 794462328877965002894838885122, 13398419999037765629218732004567606814, 1330302023374557034879527995005574743144202826
Offset: 0
G.f.: A(x) = 1 + 2*x + 14*x^2 + 682*x^3 + 236826*x^4 + 525175434*x^5 +...
such that
log(A(x))/2 = x + 12*x^2/2 + 985*x^3/3 + 470832*x^4/4 + 1311738121*x^5/5 + 21300003689580*x^6/6 + 2015874949414289041*x^7/7 +...+ Pell(n^2)*x^n/n +...
Pell numbers begin:
A000129 = [1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,...].
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{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 2*Pell(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-3 of 3 results.
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