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A213527 E.g.f.: exp( Sum_{n>=1} Fibonacci(n+1)*x^n/n ), where Fibonacci(n) = A000045(n).

%I A213527 #37 Oct 17 2023 11:28:14
%S A213527 1,1,3,13,79,603,5593,60859,760929,10743337,169068619,2934088509,
%T A213527 55666365007,1146171490387,25454016551601,606459860389867,
%U A213527 15430511490348097,417572419595723601,11975721373433624851,362832170044253318893,11579878816694604949071
%N A213527 E.g.f.: exp( Sum_{n>=1} Fibonacci(n+1)*x^n/n ), where Fibonacci(n) = A000045(n).
%H A213527 Michael Wallner, <a href="https://arxiv.org/abs/1706.07163">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017, p. 10.
%F A213527 E.g.f.: A(x) = (1/(1-x-x^2)) * exp( -Sum_{n>=1} Fibonacci(n-1)*x^n/n ).
%F A213527 a(n) = (n-1)! * Sum_{i=1..n} Fibonacci(i+1)*a(n-i)/(n-i)! with a(0)=1, where Fibonacci(n) = A000045(n). - _Vladimir Kruchinin_, Sep 29 2016
%F A213527 E.g.f.: A(x) = (1/(1-x-x^2)) * (1 - 2*x/(-1 + sqrt(5)))^((-1 + sqrt(5))/(2*sqrt(5))) * (1 + 2*x/(1 + sqrt(5)))^((1 + sqrt(5))/(2*sqrt(5))). - _Fabian Pereyra_, Oct 09 2023
%F A213527 a(n) ~ sqrt(2*Pi) * phi^(n + 1/2 - 1/(2*sqrt(5))) * n^(n + 1/(2*sqrt(5))) / (5^(1/(2*sqrt(5)*phi)) * Gamma(phi/sqrt(5)) * exp(n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Oct 10 2023
%F A213527 E.g.f.: A(x) = ( (1 + x/phi)/(1 - phi*x) )^(sqrt(5)/10) / sqrt(1-x-x^2) = exp( arctanh(sqrt(5)*x/(2-x)) / sqrt(5) ) / sqrt(1-x-x^2), where phi = A001622 is the golden ratio, from the Michael Wallner reference (corrected). - _Paul D. Hanna_, Oct 16 2023
%e A213527 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 603*x^5/5! + ...
%e A213527 such that log(A(x)) = x + 2*x^2/2 + 3*x^3/3 + 5*x^4/4 + 8*x^5/5 + 13*x^6/6 + 21*x^7/7 + 34*x^8/8 + ... + A000045(n+1)*x^n/n + ...
%t A213527 a[0] = 1; a[n_] := a[n] = (n-1)! Sum[Fibonacci[i+1] a[n-i]/(n-i)!, {i, n}];
%t A213527 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 26 2018, after _Vladimir Kruchinin_ *)
%o A213527 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n,fibonacci(m+1)*x^m/m)+x*O(x^n)),n)}
%o A213527 for(n=0,30,print1(a(n),", "))
%o A213527 (Maxima) a(n):=if n=0 then 1 else (n-1)!*sum((fib(i+1)*a(n-i))/(n-i)!,i,1,n); /* _Vladimir Kruchinin_, Sep 29 2016 */
%Y A213527 Cf. A000045.
%K A213527 nonn
%O A213527 0,3
%A A213527 _Paul D. Hanna_, Jun 13 2012