A261050 -id:A261050 - 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

A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 10, 22, 48, 104, 220, 460, 954, 1956, 3976, 8026, 16084, 32032, 63440, 124974, 245008, 478204, 929452, 1799508, 3471396, 6673724, 12788976, 24433528, 46546738, 88432264, 167575474, 316768948, 597389576, 1124092476, 2110661644, 3955006820, 7396477224

Offset: 0

Vaclav Kotesovec, Aug 18 2015

nonn

Convolution of A166861 and A261050.

G. C. Greubel, 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. A000045, A001622, A166861, A261050, A261519, A261520.

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

a(n) ~ phi^n / (2^(3/4) * 5^(1/8) * sqrt(Pi) * n^(3/4)) * exp(-1/5 + 2*5^(-1/4)*sqrt(2*n) + s), where s = 2 * Sum_{k>=1} phi^(2*k+1) / ((phi^(4*k+2) - phi^(2*k+1) - 1)*(2*k+1)) = 0.276751423987223411719438512082359840225908317... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

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

A261331 Expansion of Product_{k>=1} (1+x^k)^(A000129(k)).

Original entry on oeis.org

1, 1, 2, 7, 18, 52, 143, 396, 1083, 2971, 8087, 21981, 59533, 160857, 433467, 1165542, 3126951, 8372451, 22374172, 59684669, 158941356, 422582925, 1121814072, 2973703449, 7871754065, 20809918535, 54943916547, 144891525408, 381647503607, 1004149670985

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, A261329, A261330, A261332, A261050, A261051.

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

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)^(k+1)/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = -0.1149083344289588668149210160138124159112948627968378825745674888...

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

A261332 Expansion of Product_{k>=1} (1+x^k)^(A002203(k)).

Original entry on oeis.org

1, 2, 7, 26, 83, 278, 894, 2848, 8947, 27844, 85774, 262090, 794802, 2393874, 7165622, 21327412, 63146545, 186063052, 545783103, 1594268778, 4638773567, 13447773510, 38850645513, 111874844146, 321166890522, 919314145044, 2624198013317, 7471158542418

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, A261330, A261331, A261050, A261051.

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

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)^(k+1)/(((1+sqrt(2))^k + 2/(1 + (1+sqrt(2))^k) - 3)*k) = -0.2731939535370496116124191192900280854879921353977...

A357475 Expansion of Product_{k>=1} 1 / (1 + x^k)^Fibonacci(k).

Original entry on oeis.org

1, -1, 0, -2, 0, -3, 0, -4, 2, -5, 8, 0, 26, 19, 74, 74, 195, 221, 464, 560, 1042, 1258, 2154, 2536, 3997, 4341, 6152, 5204, 5447, -1617, -10790, -39710, -83915, -181639, -336564, -633844, -1108334, -1952371, -3293590, -5568202, -9148916, -15017471, -24144556, -38697396, -61005748, -95708150

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Convolution inverse of A261050.

Cf. A000045, A166861, A261050, A337009, A357179.

nmax = 45; CoefficientList[Series[Product[1/(1 + x^k)^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d) d Fibonacci[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 45}]

a(n) = Sum_{k=0..n} (-1)^k * A337009(n,k). - Alois P. Heinz, Apr 30 2023

A357179 Expansion of Product_{k>=1} (1 - x^k)^Fibonacci(k).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 2, 1, 5, 6, 14, 15, 32, 40, 64, 86, 131, 166, 237, 287, 362, 389, 368, 149, -339, -1477, -3680, -7827, -15245, -28270, -50493, -87886, -149827, -250966, -414542, -675741, -1089267, -1736640, -2741788, -4284837, -6632751, -10162683, -15412613, -23110653, -34236290

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Convolution inverse of A166861.

Cf. A000045, A166861, A261050, A357475.

nmax = 45; CoefficientList[Series[Product[(1 - x^k)^Fibonacci[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d Fibonacci[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 45}]

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).

cp's OEIS Frontend

A166861 Euler transform of Fibonacci numbers.

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A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

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

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A261331 Expansion of Product_{k>=1} (1+x^k)^(A000129(k)).

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A261332 Expansion of Product_{k>=1} (1+x^k)^(A002203(k)).

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A357475 Expansion of Product_{k>=1} 1 / (1 + x^k)^Fibonacci(k).

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A357179 Expansion of Product_{k>=1} (1 - x^k)^Fibonacci(k).

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

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