A166861 -id:A166861 - cp's OEIS frontend

A261329 Euler transform of Pell numbers.

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

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)])

A111335 Let qf(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is qf(q^3,q^4)/qf(q,q^4).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, -1, 0, 2, 1, -1, -1, 1, 2, -1, -2, 1, 3, 1, -3, -1, 3, 1, -3, -2, 4, 4, -3, -4, 3, 5, -3, -7, 2, 9, 0, -9, -1, 10, 3, -11, -5, 12, 8, -11, -10, 10, 12, -11, -15, 11, 19, -7, -21, 6, 24, -5, -28, 1, 31, 4, -33, -8, 36, 12, -38, -18, 40, 27, -40, -33, 40, 39, -40, -49, 38, 60, -34, -67, 30, 75, -25, -87, 18, 98

Offset: 0

N. J. A. Sloane, Nov 09 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A111330.

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, -1)):
seq(a(n), n=0..86); # Peter Luschny, Nov 17 2022

{a(n)=if(n<0, 0, polcoeff( prod(k=0,n\2, (1-x^(2*k+1))^(-(-1)^k), 1+x*O(x^n)), n))} /* Michael Somos, Nov 11 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, -1)
a = EulerTransform(b)
print([a(n) for n in range(87)]) # Peter Luschny, Nov 17 2022

Euler transform of period 4 sequence [1, 0, -1, 0, ...]. - Michael Somos, Nov 10 2005
G.f.: Product_{k>0} (1-x^(2k-1))^((-1)^k). - Michael Somos, Nov 11 2005
G.f.: exp( Sum_{k >= 1} 1/(k*(x^k + x^(-k))) ). - Peter Bala, Sep 28 2023

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

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

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

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...

A292386 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -3, -3, -1, 10, 20, 36, 28, -11, -103, -245, -397, -448, -214, 464, 1817, 3680, 5660, 6473, 4362, -3232, -18428, -41946, -70589, -94890, -96996, -49673, 78907, 317995, 673299, 1105044, 1491333, 1605102, 1094914, -479358, -3561322, -8404118, -14781724, -21595744, -26450603, -25329527

Offset: 0

Ilya Gutkovskiy, Sep 15 2017

sign

Convolution inverse of A000294 (Euler transform of the triangular numbers).

Cf. A000294, A027999, A028377.

nmax = 41; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

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

A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 28, 47, 88, 152, 279, 486, 876, 1539, 2744, 4824, 8551, 15023, 26503, 46509, 81747, 143210, 251007, 438915, 767403, 1339487, 2336955, 4071906, 7090589, 12333894, 21440241, 37235775, 64624267, 112067176, 194209732, 336313393, 582019000

Offset: 0

Thomas Wieder, Jun 30 2005

nonn

A109509 is the Euler transform of the right-shifted Fibonacci numbers A000045.

			Let * denote an unlabeled element.
Let | denote a delimiter between two hierarchies. E.g., for n=3 we have in **|* two hierarchies (each with one level only).
Let : denote a higher level (within a single hierarchy). E.g., for n=6 we have in ***:**:* a single hierarchy distributed over three levels.
Then a(5) = 4 because we have *****, ***:**, **:***, **|***.

Alois P. Heinz, 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.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A000045, A075729, A097236, A166861.

SeqSetSetxU := [T, {T=Set(S),S=Sequence(U,card>=1),U=Set(Z,card>=2)},unlabeled]; seq(count(SeqSetSetxU,size=j),j=1..25); # where x is an integer 1, 2, 3,... # x=2 gives 2 individuals per level.

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k-1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 06 2015 *)

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

a(n) ~ phi^(n-1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 1/2 + 2*5^(-1/4)*sqrt(n/phi) + s), where s = Sum_{k>=2} 1/((phi^(2*k) - phi^k - 1)*k) = 0.189744799982532613329750744326543900883761701983311537716143669... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015

Edited with more terms from Franklin T. Adams-Watters, Oct 21 2009

A117410 Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.

Original entry on oeis.org

1, 1, -1, 0, -1, -2, 1, -1, -1, 0, 1, 1, -1, 1, 0, 2, 1, 0, 0, -1, 2, 1, 0, -1, 0, -1, 0, -1, 1, 1, -3, 0, -1, -1, -1, 1, 0, 0, 0, -1, -2, 0, 1, 0, 1, 0, 1, 0, 0, -1, 2, -1, 0, 1, 1, 3, 0, -1, 0, 1, -1, 0, 1, 0, 0, 2, 0, 1, -1, 0, -2, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, -1, -1, -1, 0, -2, -1, 0, 2, 1, -2, 0, 1, -1, 0, -2, -1, 1, -1, 1, 0, 0, 0, 1, 0

Offset: 0

Michael Somos, Mar 13 2006

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x - x^2 - x^4 - 2*x^5 + x^6 - x^7 - x^8 + x^10 + x^11 - x^12 + x^13 + ...
G.f. = q^5 + q^29 - q^53 - q^101 - 2*q^125 + q^149 - q^173 - q^197 + q^245 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 183. MR1019331 (90k:11050).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A107034.

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, 1, -2)):
seq(a(n), n = 0..104); # Peter Luschny, Nov 17 2022

a[ n_] := SeriesCoefficient[ QPochhammer[x^2]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A), n))};

q='q+O('q^99); Vec(eta(q^2)^3/eta(q)) \\ Altug Alkan, Apr 17 2018

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, -2)
a = EulerTransform(b)
print([a(n) for n in range(105)]) # Peter Luschny, Nov 17 2022

Expansion of psi(x)^2 * chi(-x) = f(-x)^2 / chi(-x)^3 = f(-x)^5 / phi(-x)^3 = f(-x^2)^2 / chi(-x) = f(-x^2)^3 / f(-x) = psi(x) * f(-x^2) = f(x) * f(-x^4) = phi(-x)^2 / chi(-x)^5 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 31 2015
Euler transform of period 2 sequence [ 1, -2, ...].
Given A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A3 * A1^2 - A2 * A4^2.
Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A6 + A1*A5 + A2*A4 + 4*A3^2, A3 = x^10 * A(x^49).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2.
A107034(n) = (-1)^n * a(n).

A347011 Euler transform of j-> ceiling(2^(j-2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 999, 2185, 4796, 10470, 22871, 49815, 108427, 235515, 511074, 1107248, 2396299, 5179169, 11181877, 24113939, 51949572, 111801422, 240381703, 516355235, 1108186951, 2376314763, 5091422730, 10900063776, 23317805916

Offset: 0

Alois P. Heinz, Aug 10 2021

nonn

Differs from A206301 first at n=10.

Alois P. Heinz, Table of n, a(n) for n = 0..3250

Cf. A034691, A034899, A166861, A187251, A206301, A319918.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, i-1)*binomial(ceil(2^(i-2))+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
       ceil(2^(d-2)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(1-x) * Product[1/(1 - x^k)^(2^(k-2)), {k, 2, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 11 2021 *)

G.f.: Product_{j>0} 1/(1-x^j)^ceiling(2^(j-2)).

cp's OEIS Frontend

A261329 Euler transform of Pell 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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A111335 Let qf(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is qf(q^3,q^4)/qf(q,q^4).

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

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A261330 Euler transform of Pell-Lucas numbers.

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A292386 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

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A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

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