A261031 -id:A261031 - cp's OEIS frontend

A166861 Euler transform of Fibonacci numbers.

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

Offset: 0

Franklin T. Adams-Watters, Oct 21 2009

nonn

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

			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 + ...

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

Cf. A000045, A034691, A109509, A200544, A260787, A261031, A261050, A260916, row sums of A337009.

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

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 05 2015 *)

ET(v)=Vec(prod(k=1,#v,1/(1-x^k+x*O(x^#v))^v[k]))
ET(vector(40,n,fibonacci(n)))

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

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

First formula corrected by Vaclav Kotesovec, Aug 05 2015

A261329 Euler transform of Pell numbers.

Original entry on oeis.org

1, 1, 3, 8, 23, 62, 175, 477, 1319, 3602, 9851, 26779, 72726, 196724, 531157, 1430144, 3842911, 10303055, 27570786, 73637306, 196333303, 522584286, 1388786089, 3685169795, 9764703347, 25838430572, 68282175170, 180221449469, 475102410065, 1251038486529

Offset: 0

Vaclav Kotesovec, Aug 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.
Eric Weisstein's World of Mathematics, Pell Number
Wikipedia, Pell number

Cf. A000129, A261330, A261331, A261332, A166861, A261031.

nmax=40; Pell[0]=0; Pell[1]=1; Pell[n_]:=Pell[n] = 2*Pell[n-1] + Pell[n-2]; CoefficientList[Series[Product[1/(1-x^k)^Pell[k], {k, 1, nmax}], {x, 0, nmax}], x]

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 1)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k>=1} 1/(1-x^k)^(A000129(k)).
a(n) ~ (1+sqrt(2))^n * exp(-1/8 + 2^(1/4)*sqrt(n) + s) / (2^(11/8) * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} 1/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = 0.17615706029370539578355193664752741450665073523628663099586621933373...
G.f.: exp(Sum_{k>=1} x^k/(k*(1 - 2*x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A358369 Euler transform of 2^floor(n/2), (A016116).

Original entry on oeis.org

1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636

Offset: 0

Peter Luschny, Nov 17 2022

nonn

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;
u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;
b*u(n - 1) + c*u(n - 2) end; u end:
EulerTransform := proc(a) local b;
b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),
d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:
a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

from typing import Callable
from functools import cache
from sympy import divisors
def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:
    @cache
    def u(n: int) -> int:
        if n < 2:
            return [u0, u1][n]
        return b * u(n - 1) + c * u(n - 2)
    return u
def EulerTransform(a: Callable) -> Callable:
    @cache
    def b(n: int) -> int:
        if n == 0:
            return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)
            for j in range(1, n + 1))
        return s // n
    return b
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

Original entry on oeis.org

1, 1, 3, 7, 14, 33, 69, 148, 307, 642, 1314, 2684, 5432, 10924, 21841, 43431, 85913, 169170, 331675, 647601, 1259737, 2441706, 4716874, 9083215, 17439308, 33387589, 63749174, 121409236, 230658963, 437198116, 826838637, 1560410267, 2938808875, 5524005110

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Cf. A000032, A261031, A261050, A026007, A027998, A248882, A102866, A256142.

L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(L(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^LucasL[k],{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = -0.590290697526802161885355317939144642488927381134222996704542... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A261330 Euler transform of Pell-Lucas numbers.

Original entry on oeis.org

1, 2, 9, 30, 106, 348, 1153, 3698, 11798, 37034, 115294, 355202, 1086080, 3294912, 9931019, 29745296, 88597104, 262508288, 774073787, 2272321666, 6642701371, 19342768210, 56117550874, 162247236638, 467563212923, 1343273262184, 3847866714452, 10991864363660

Offset: 0

Vaclav Kotesovec, Aug 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Eric Weisstein's World of Mathematics, Pell Number
Wikipedia, Pell number

Cf. A002203, A261329, A261331, A261332, A166861, A261031.

nmax=40; cPell[0]=2; cPell[1]=2; cPell[n_]:=cPell[n] = 2*cPell[n-1] + cPell[n-2]; CoefficientList[Series[Product[1/(1-x^k)^cPell[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1-x^k)^(A002203(k)).

a(n) ~ (1+sqrt(2))^n * exp(-1 + 2^(-3/2) + 2*sqrt(n) + s) / (2 * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} = 2/(((1+sqrt(2))^k + 2/(1 + (1+sqrt(2))^k) - 3)*k) = 0.40371233206538058741995064489690066306587648488344483...

A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204.

Original entry on oeis.org

1, 1, 4, 8, 24, 47, 129, 255, 641, 1308, 3064, 6225, 14286, 28792, 63571, 129240, 278329, 561044, 1190501, 2387695, 4987250, 9976529, 20536591, 40879937, 83416195, 165182927, 333581057, 658385847, 1318764282, 2590568669, 5154370637, 10082762399, 19929958391, 38848175389, 76331335061, 148233818041

Offset: 0

Ilya Gutkovskiy, Feb 18 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000

Cf. A000204, A261031, A300520.

nmax = 35; CoefficientList[Series[Product[1/(1 - LucasL[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[Sum[LucasL[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d LucasL[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} Lucas(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 23 2019: (Start)
a(n) ~ c * 3^(n/2), where
c = 27050904.849254721356174679220734831574107371522481898944915... if n is even,
c = 27050894.152054775323471273913497954429537332266942696921416... if n is odd.
In closed form, c = ((3 + sqrt(3)) * Product_{k>=3}(1/(1 - Lucas(k)/3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3}(1/(1 - (-1)^k*Lucas(k)/3^(k/2))))/4.
(End)

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A166861 Euler transform of Fibonacci numbers.

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A261329 Euler transform of Pell numbers.

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A358369 Euler transform of 2^floor(n/2), (A016116).

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A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

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A261330 Euler transform of Pell-Lucas numbers.

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A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204.

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