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

1, 1, 3, 13, 79, 603, 5593, 60859, 760929, 10743337, 169068619, 2934088509, 55666365007, 1146171490387, 25454016551601, 606459860389867, 15430511490348097, 417572419595723601, 11975721373433624851, 362832170044253318893, 11579878816694604949071

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 10.

Formula

E.g.f.: A(x) = (1/(1-x-x^2)) * exp( -Sum_{n>=1} Fibonacci(n-1)*x^n/n ).

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

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

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

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

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 603*x^5/5! + ...
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 + ...

Mathematica

a[0] = 1; a[n_] := a[n] = (n-1)! Sum[Fibonacci[i+1] a[n-i]/(n-i)!, {i, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 26 2018, after Vladimir Kruchinin *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n, fibonacci(m+1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(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 */

Crossrefs

Cf. A000045.

Sequence in context: A215915 A159312 A355165 * A261601 A125659 A010844

Adjacent sequences: A213524 A213525 A213526 * A213528 A213529 A213530

Keyword

nonn

Author

Paul D. Hanna, Jun 13 2012

Status

approved