A084099 -id:A084099 - cp's OEIS frontend

A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

0, 1, -2, 12, -110, 1380, -22022, 426972, -9747950, 256176660, -7617417302, 252851339532, -9268406209790, 371843710214340, -16206868062692582, 762569209601624892, -38525315595630383630, 2079964082064837282420, -119513562475103977951862

Offset: 0

Michael De Vlieger, May 16 2025

The series -postLie(-x) is the inverse for the substitution of the series comTrias(x), given by the suspension of the Koszul dual of comTrias. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..374
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, commutative triassociative operad "ComTrias".

Cf. A002050, A006531, A084099, A097388, A101851, A114285, A225883, A383985, A383986, A383987, A383989. Composition of -A006963(-x) and exp(x)-1.

nn = 18; f[x_] := Log[(1 + Sqrt[1 + 4*x])/2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A084100 Expansion of (1+x-x^2-x^3)/(1+x^2).

Original entry on oeis.org

1, 1, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 2, -2

Offset: 0

Paul Barry, May 15 2003

Partial sums are A084099.

The unsigned sequence 1,1,2,2,2,2,.. has g.f. (1+x^2)/(1-x) and a(n)=sum{k=0..n, binomial(1,k/2)(1+(-1)^k)/2}. Its partial sums are A004275(n+1). The sequence 1,-1,2,-2,2,-2,... has g.f. (1+x^2)/(1+x) and a(n)=sum{k=0..n, (-1)^(n-k)binomial(1,k/2)(1+(-1)^k)/2}. - Paul Barry, Oct 15 2004

			G.f. = 1 + x - 2*x^2 - 2*x^3 + 2*x^4 + 2*x^5 - 2*x^6 - 2*x^7 + 2*x^8 + 2*x^9 + ...

Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A084099, A280560.

CoefficientList[Series[(1+x-x^2-x^3)/(1+x^2),{x,0,100}],x]  (* Harvey P. Dale, Apr 20 2011 *)
a[ n_] := (-1)^Quotient[n, 2] If[ Quotient[n, 2] != 0, 2, 1]; (* Michael Somos, Jan 05 2017 *)

{a(n) = (-1)^(n\2) * if( n\2, 2, 1)}; /* Michael Somos, Jan 05 2017 */

Euler transform of length 4 sequence [1, -3, 0, 1]. - Michael Somos, Jan 05 2017
G.f.: (1 + x) * (1 - x^2) / (1 + x^2). - Michael Somos, Jan 05 2017
a(n) = a(1-n) for all n in Z. - Michael Somos, Jan 05 2017
a(2*n) = a(2*n + 1) = A280560(n) for all n in Z. - Michael Somos, Jan 05 2017

A100088 Expansion of (1-x^2)/((1-2x)(1+x^2)).

Original entry on oeis.org

1, 2, 2, 4, 10, 20, 38, 76, 154, 308, 614, 1228, 2458, 4916, 9830, 19660, 39322, 78644, 157286, 314572, 629146, 1258292, 2516582, 5033164, 10066330, 20132660, 40265318, 80530636, 161061274, 322122548, 644245094, 1288490188, 2576980378

Offset: 0

Paul Barry, Nov 03 2004

A Chebyshev transform of A100087, under the mapping A(x) -> ((1-x^2)/(1+x^2)) * A(x/(1+x^2)).

A176742(n+2) = A084099(n+2) = period 4:repeat 0, -2, 0, 2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2).

Cf. A007910, A084099, A100087, A122117, A176742.

[n le 3 select Floor((n+2)/2) else 2*Self(n-1) - Self(n-2) +2*Self(n-3): n in [1..41]]; // G. C. Greubel, Jul 08 2022

CoefficientList[Series[(1-x^2)/((1-2x)(1+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2},{1,2,2},40] (* Harvey P. Dale, May 12 2011 *)

def A100088(n): return ((4<Chai Wah Wu, Apr 22 2025

def b(n): return (2/5)*(3*2^(2*n-1) + (-1)^n) # b=A122117
def A100088(n): return b(n/2) if (n%2==0) else 2*b((n-1)/2)
[A100088(n) for n in (0..60)]  # G. C. Greubel, Jul 08 2022

a(n) = (3*2^n + 2*cos(Pi*n/2) + 4*sin(Pi*n/2))/5.
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A100087(n-2*k)/(n-k).
a(n) = 2*a(n-1) + period 4:repeat 0, -2, 0, 2, with a(0) = 1.
a(n) = A007910(n+1) - A007910(n-1).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = (1/5)*(3*2^n + i^n*(1+(-1)^n) - 2*i^(n+1)*(1-(-1)^n)). - G. C. Greubel, Jul 08 2022
a(n) = A122117(n/2) if (n mod 2 = 0) otherwise 2*A122117((n-1)/2). - G. C. Greubel, Jul 21 2022

cp's OEIS Frontend

A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

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A084100 Expansion of (1+x-x^2-x^3)/(1+x^2).

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A100088 Expansion of (1-x^2)/((1-2x)(1+x^2)).

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cp's OEIS Frontend

A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

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Mathematica

A084100 Expansion of (1+x-x^2-x^3)/(1+x^2).

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Author

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Programs

Mathematica

PARI

Formula

A100088 Expansion of (1-x^2)/((1-2*x)*(1+x^2)).

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Magma

Mathematica

Python

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A100088 Expansion of (1-x^2)/((1-2x)(1+x^2)).