A207972
Expansion of g.f.: exp( Sum_{n>=1} 5*Fibonacci(n^2) * x^n/n ).
Original entry on oeis.org
1, 5, 20, 115, 1665, 82650, 12847310, 5620114060, 6659421195205, 21082748688390045, 177217804775828062850, 3941798437750184226876305, 231505293200405380457355524620, 35848160499603817968830380832049915, 14619744406297572472084577939841875791890
Offset: 0
G.f.: A(x) = 1 + 5*x + 20*x^2 + 115*x^3 + 1665*x^4 + 82650*x^5 + ...
such that
log(A(x))/5 = x + 3*x^2/2 + 34*x^3/3 + 987*x^4/4 + 75025*x^5/5 + 14930352*x^6/6 + 7778742049*x^7/7 + ... + Fibonacci(n^2)*x^n/n + ...
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{a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,16,print1(a(n),", "))
A211893
G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^n * x^n/n ), where Jacobsthal(n) = A001045(n).
Original entry on oeis.org
1, 3, 6, 36, 561, 98211, 43176384, 116622937722, 1022189210900601, 41675008108242048327, 6377839090284322052067558, 4114890941608928235401688095580, 10460015732506081308723488849683574907, 108482611110966450613465001912856742180485969
Offset: 0
G.f.: A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 561*x^4 + 98211*x^5 + 43176384*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^3*x^3/3 + 5^4*x^4/4 + 11^5*x^5/5 + 21^6*x^6/6 + 43^7*x^7/7 +...+ Jacobsthal(n)^n*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].
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{Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^k*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))
A231279
a(n) = Jacobsthal(n^2), where Jacobsthal(n) = A001045(n), for n>=1.
Original entry on oeis.org
1, 5, 171, 21845, 11184811, 22906492245, 187649984473771, 6148914691236517205, 805950546409752783137451, 422550200076076467165567735125, 886151997189943915269204706853563051, 7433581732843541047178572757549453835326805
Offset: 1
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{a(n)=(2^(n^2)-(-1)^n)/3}
for(n=1,15,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 1-4 of 4 results.
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