A261329 -id:A261329 - cp's OEIS frontend

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

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)])

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...

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...

A308448 Expansion of Sum_{k>=1} mu(k)log(1 + x^k/(1 - 2x^k - x^(2*k)))/k.

Original entry on oeis.org

1, 1, 3, 6, 14, 28, 64, 135, 300, 653, 1458, 3223, 7240, 16228, 36678, 83025, 188910, 430730, 985752, 2260866, 5199612, 11982591, 27673826, 64027215, 148399514, 344490100, 800886300, 1864461210, 4346031950, 10142519585, 23696421808, 55420499295, 129742683174, 304014091125

Offset: 1

Ilya Gutkovskiy, May 27 2019

nonn

Inverse Euler transform of A000129.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000045, A000129, A006206, A008683, A060280, A261329.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> combinat[fibonacci](n)+b(n, n-1):
seq(a(n), n=1..34);  # Alois P. Heinz, May 19 2022

nmax = 34; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - 2 x^k - x^(2 k))]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 40; s = ConstantArray[0, nmax]; Do[s[[j]] = j*Fibonacci[j, 2] - Sum[s[[d]]*Fibonacci[j - d, 2], {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 10 2019 *)

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000129.
a(n) ~ (1 + sqrt(2))^n/n. - Vaclav Kotesovec, May 28 2019
"CHK" (necklace, identity, unlabeled) transform of A000045. - Alois P. Heinz, May 19 2022

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

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A261330 Euler transform of Pell-Lucas numbers.

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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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A308448 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - 2*x^k - x^(2*k)))/k.

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A308448 Expansion of Sum_{k>=1} mu(k)log(1 + x^k/(1 - 2x^k - x^(2*k)))/k.