A383991 -id:A383991 - cp's OEIS frontend

A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

0, 1, -11, 61, -215, -1559, 62941, -1371131, 26310481, -474554735, 7824076741, -98881279859, -176260664711, 87457412423161, -5077434546358355, 234510433823788501, -10016559114085864799, 413333665704129673249, -16704968283664639137899, 660340818239784197391325

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series ff6^!(x) is the inverse for the substitution of the series ff6(x) (given by A231690), given by the suspension of the Koszul dual of FF6. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..393
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad "FF6".

Cf. A003725, A097388, A111884, A112242, A177885, A318215, A383989, A383990, A383991, A383992, A383993, A383994.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3)], {x, 0, nn}], x]

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -725, 14401, -360005, 10863889, -384415925, 15612336481, -715930020005, 36592369889329, -2062911091119125, 127170577711282561, -8510569547826528005, 614491222512504748369, -47615614242877583230325, 3941408640018910366196641

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..353
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, triassociative operad "Trias".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383988, A383989, A383991.

Composition of A001003 with exp(x)-1.

nn = 17; f[x_] := (1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x); Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

Original entry on oeis.org

0, 1, -4, 3, 40, -330, 1626, -3150, -54592, 1060920, -13022280, 127171440, -889086648, -283184616, 179750627616, -4895777544840, 99124001788800, -1721513264431680, 25736021675994816, -292896125040673728, 639149345262276480, 106178474282318726400

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..448
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad "Arbustive".

Cf. A003725, A097388, A111884, A112242, A114285, A177885, A318215, A383990, A383991, A383993, A383994, A383995.

nn = 21; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(Log[1 + x] - x^2)/(1 + x)], {x, 0, nn}], x]

A383993 Series expansion of the exponential generating function exp(tridup^!(x)) - 1 where tridup^!(x) = x / ((1+x) * (1+2*x)).

Original entry on oeis.org

0, 1, -5, 25, -119, 301, 5611, -171275, 3574705, -68597639, 1282415131, -23479249199, 409082338105, -6146707844315, 46462772999371, 2072826643602541, -160983324879816479, 8004468391727017585, -352443295329194182085, 14817357881274444545161

Offset: 0

Michael De Vlieger, May 16 2025

The series tridup^!(x) is the inverse for the substitution of the series tridup(x) (given by A001003), given by the suspension of the Koszul dual of tridup. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..404
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, triduplicial operad "TriDup".

Cf. A002050, A003725, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383994, A383995.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x/((1 + x)*(1 + 2*x))], {x, 0, nn}], x]

A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

Original entry on oeis.org

0, 1, -2, 0, 12, -60, 240, -840, 1680, 15120, -332640, 4656960, -59209920, 735134400, -9098369280, 112345833600, -1365274310400, 15746578848000, -155630893017600, 762963647846400, 22567767443020800, -1126188650069683200, 35900904478389350400

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series wnp^!(x) is the inverse for the substitution of the series wnp(x) (corresponding to A048172), given by the suspension of the Koszul dual of the WithoutNPosets operad. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..451
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad without n posets "WNP".

Cf. A003725, A084099, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383993, A383995.

nn = 22; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[Log[1 + x] - x^2/(1 + x)], {x, 0, nn}], x]

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

Original entry on oeis.org

0, 1, -3, 19, -191, 2661, -47579, 1040047, -26888511, 802727209, -27178685459, 1029077910411, -43086906080063, 1976633329627789, -98597207392040811, 5313105048925173991, -307587436319162110079, 19038773384213189214417, -1254686724727364725716131

Offset: 0

Michael De Vlieger, May 16 2025

sign

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

Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, diassociative operad "Dias".

Cf. A003725, A006531, A097388, A111884, A112242, A177885, A318215, A383991, A383992, A383993, A383994, A383995. Composition of exp(x)-1 with -A000108(-x).

cp's OEIS Frontend

A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3*x - sqrt(1+6*x+x^2)) / (4*x) (A001003).

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A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

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A383993 Series expansion of the exponential generating function exp(tridup^!(x)) - 1 where tridup^!(x) = x / ((1+x) * (1+2*x)).

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A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

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Crossrefs

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Mathematica

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4*x)) / (2*x) + 1 (given by A000108).

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A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).