A200544 -id:A200544 - 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

A337009 Triangle of the Multiset Transform of the Fibonacci Sequence.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 8, 11, 6, 3, 1, 1, 13, 19, 13, 6, 3, 1, 1, 21, 37, 25, 14, 6, 3, 1, 1, 34, 65, 52, 27, 14, 6, 3, 1, 1, 55, 120, 98, 58, 28, 14, 6, 3, 1, 1, 89, 210, 191, 113, 60, 28, 14, 6, 3, 1, 1, 144, 376, 360, 229, 119, 61, 28, 14, 6, 3, 1, 1, 233, 654, 678, 443, 244, 121, 61, 28, 14, 6, 3, 1, 1

Offset: 1

R. J. Mathar, Aug 11 2020

Short definition of the Multiset Transformation: supposed we have F(w) distinct objects of weight w. Then T(n,k) is the number of bags of objects with total weight n containing k objects. Multisets means that objects may appear more than once in the bag, but the order of the objects in the bag does not matter.

Apparently A200544 is the limit of the reversed rows as n approaches infinity.

			The triangle starts with rows n>=1 and columns k>=1:
    1
    1     1
    2     1     1
    3     3     1     1
    5     5     3     1     1
    8    11     6     3     1     1
   13    19    13     6     3     1     1
   21    37    25    14     6     3     1     1
   34    65    52    27    14     6     3     1     1
   55   120    98    58    28    14     6     3     1     1
   89   210   191   113    60    28    14     6     3     1     1
  144   376   360   229   119    61    28    14     6     3     1     1
  233   654   678   443   244   121    61    28    14     6     3     1     1
  377  1149  1255   866   481   250   122    61    28    14     6     3     1  1
  ...

Alois P. Heinz, Rows n = 1..200, flattened
Index to Sequence pairs related by Multiset transformations

Cf. A000045 (column k=1), A089098 (column k=2), A166861 (row sums), A200544 (limiting row?), A357475.

F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
      add(binomial(F(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 29 2021

nn = 13;
Rest@CoefficientList[#, y]& /@ (Series[Product[1/(1 - y x^i)^Fibonacci[i], {i, 1, nn}], {x, 0, nn}] // Rest@CoefficientList[#, x]&) // Flatten (* Jean-François Alcover, Oct 29 2021 *)

G.f.: Product_{j>=1} 1/(1-y*x^j)^Fibonacci(j). - Jean-François Alcover, Oct 29 2021

Sum_{k=0..n} (-1)^k * T(n,k) = A357475(n). - Alois P. Heinz, Apr 30 2023

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)])

A260787 G.f.: Product_{k>=1} 1/(1-x^k)^Fibonacci(k+2).

Original entry on oeis.org

1, 2, 6, 15, 38, 89, 210, 474, 1065, 2339, 5091, 10919, 23230, 48887, 102126, 211599, 435561, 890617, 1810786, 3661118, 7365473, 14747049, 29397160, 58356179, 115392801, 227332038, 446304671, 873298579, 1703463864, 3312873935, 6424553973, 12425158365, 23968214357, 46120280910, 88535346223

Offset: 0

N. J. A. Sloane, Aug 05 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4450
W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014. See F_H.
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, A166861, A200544.

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k+2], {k, 1, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 05 2015 *)

a(n) ~ phi^(n+1/2) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 7/10 + 2*5^(-1/4)*phi*sqrt(n) + s), where s = Sum_{k>=2} (1 + 2*phi^k) / ((phi^(2*k) - phi^k - 1)*k) = 1.39069800276768443926918973402733105305129194986259856042723... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015

A291650 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))^Fibonacci(k).

Original entry on oeis.org

1, 1, 2, 5, 4, 12, 14, 16, 42, 35, 65, 100, 84, 205, 201, 254, 490, 386, 749, 917, 851, 1816, 1566, 2260, 3513, 2784, 5566, 5748, 6116, 11366, 9048, 14740, 19037, 16095, 31576, 28505, 35218, 56334, 43671, 77512, 85163, 80577, 147756, 121016, 172408, 236022

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct Fibonacci parts (1 counted as single Fibonacci number), where Fibonacci(k) different parts of size Fibonacci(k) are available (1a, 2a, 2b, 3a, 3b, 3c, ...).

			a(3) = 5 because we have [3a], [3b], [3c], [2a, 1a] and [2b, 1a].

Index entries for related partition-counting sequences

Cf. A000045, A000119, A000121, A026007, A166861, A200544, A239002, A260787, A261050.

nmax = 50; CoefficientList[Series[Product[(1 + x^Fibonacci[k])^Fibonacci[k], {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A000045(k))^A000045(k).

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A166861 Euler transform of Fibonacci numbers.

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A337009 Triangle of the Multiset Transform of the Fibonacci Sequence.

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

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A260787 G.f.: Product_{k>=1} 1/(1-x^k)^Fibonacci(k+2).

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A291650 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))^Fibonacci(k).

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