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 A166861 Euler transform of Fibonacci numbers. 33
 1, 1, 2, 4, 8, 15, 30, 56, 108, 203, 384, 716, 1342, 2487, 4614, 8510, 15675, 28749, 52652, 96102, 175110, 318240, 577328, 1045068, 1888581, 3406455, 6134530, 11029036, 19799363, 35490823, 63531134, 113570988, 202767037, 361565865, 643970774, 1145636750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS In general, the sequence with g.f. Product_{k>=1} 1/(1-x^k)^Fibonacci(k+z), where z is nonnegative integer, is asymptotic to phi^(n + z/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp((phi/10 - 1/2) * Fibonacci(z) - Fibonacci(z+1)/10 + 2 * 5^(-1/4) * phi^(z/2) * sqrt(n) + s), where s = Sum_{k>=2} (Fibonacci(z) + Fibonacci(z+1) * phi^k) / ((phi^(2*k) - phi^k - 1)*k) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..4550 Loic Foissy, The Hopf algebra of Fliess operators and its dual pre-Lie algebra, 2013. W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014. Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015 Vaclav Kotesovec, Asymptotics of sequence A034691 FORMULA G.f.: Product_{k>0} 1/(1 - x^k)^Fibonacci(k). a(n) ~ phi^n / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(-1/10 + 2*5^(-1/4)*sqrt(n) + s), where s = Sum_{k>=2} phi^k / ((phi^(2*k) - phi^k - 1)*k) = 0.600476601392575912969719494850393576083765123939643511355547131467... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015 G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 29 2018 EXAMPLE G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 30*x^6 + 56*x^7 + 108*x^8 + 203*x^9 + ... MAPLE F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end: a:= proc(n) option remember; `if`(n=0, 1, add(add(d*       F(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)     end: seq(a(n), n=0..40);  # Alois P. Heinz, Jan 12 2017 MATHEMATICA CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 05 2015 *) PROG (PARI) ET(v)=Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k])) ET(vector(40, n, fibonacci(n))) (SageMath) def EulerTransform(a):     @cached_function     def b(n):         if n == 0: return 1         s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))         return s//n     return b a = BinaryRecurrenceSequence(1, 1) b = EulerTransform(a) print([b(n) for n in range(36)]) # Peter Luschny, Nov 11 2020 CROSSREFS Cf. A000045, A034691, A109509, A200544, A260787, A261031, A261050, A260916, row sums of A337009. Sequence in context: A301480 A217777 A034338 * A026023 A077596 A091865 Adjacent sequences:  A166858 A166859 A166860 * A166862 A166863 A166864 KEYWORD nonn AUTHOR Franklin T. Adams-Watters, Oct 21 2009 EXTENSIONS First formula corrected by Vaclav Kotesovec, Aug 05 2015 STATUS approved

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Last modified January 19 20:17 EST 2021. Contains 340269 sequences. (Running on oeis4.)