A101851 -id:A101851 - cp's OEIS frontend

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).

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]

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -467, 4381, -49091, 643021, -9615827, 161844541, -3026079971, 62243374381, -1396619164787, 33949401567901, -888725861445251, 24926889744928141, -745755560487363347, 23705772035082494461, -797875590555470224931, 28346366547928396344301

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..407
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, operad "FF6".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383987, A383988, A383995.

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

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

Original entry on oeis.org

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
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, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.

Original entry on oeis.org

0, 1, -1, 1, -13, 61, -601, 5881, -73333, 1021861, -16334401, 290146561, -5707536253, 122821558861, -2873553719401, 72586328036041, -1969306486088773, 57106504958139061, -1762735601974347601, 57705363524117482321, -1996916624448159410893

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..409
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, operad "NAC_2".

Cf. A002050, A006531, A084099, A101851, A114285, A182037, A225883, A383985, A383987, A383988, A383989.

nn = 20; f[x_] := -1 + Sqrt[1 + 2 x - x^2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

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).

Original entry on oeis.org

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]

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

Original entry on oeis.org

0, 0, 1, 0, -5, -10, 16, 154, 365, -750, -9749, -35222, 20956, 1013220, 6007821, 10272092, -129948837, -1405396426, -6318145964, 7407235766, 371429230721, 3172609248526, 11070816858267, -73488239926510, -1500342260080360, -11917913896465720, -31231507292803479

Offset: 0

Ilya Gutkovskiy, May 07 2024

sign

Cf. A000217, A000225, A000587, A003128, A059606, A081047, A101851.

nmax = 26; CoefficientList[Series[Exp[1 - Exp[x]] (Exp[x] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] Binomial[k, 2], {k, 0, n}], {n, 0, 26}]

a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * binomial(k,2).

a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,2) * A000587(n-k).

A372803 Expansion of e.g.f. exp(1 - exp(-x)) * (exp(-x) - 1) * (exp(-x) - 2).

Original entry on oeis.org

0, 1, 3, -2, -11, 31, 14, -349, 1047, 820, -21265, 90355, -26352, -2086083, 14092615, -32449650, -241320287, 3080629195, -15455723498, -2456654665, 760213889483, -7097893818852, 28459679925187, 125560349169887, -3153253543188992, 26852335900600041, -86130449768002245

Offset: 0

Ilya Gutkovskiy, May 14 2024

sign

Cf. A000587, A033452, A045406, A101851, A151881.

nmax = 26; CoefficientList[Series[Exp[1 - Exp[-x]] (Exp[-x] - 1) (Exp[-x] - 2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS2[n, k] k^2, {k, 0, n}], {n, 0, 26}]

G.f.: Sum_{k>=0} k^2 * x^k / Product_{j=0..k} (1 + j*x).

a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * k^2.

cp's OEIS Frontend

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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A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

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A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

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A383986 Expansion of the exponential generating function sqrt(4*exp(x) - exp(2*x) - 2) - 1.

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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

A372624 Expansion of e.g.f. exp(1 - exp(x)) * (exp(x) - 1)^2 / 2.

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Formula

A372803 Expansion of e.g.f. exp(1 - exp(-x)) * (exp(-x) - 1) * (exp(-x) - 2).

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Formula

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).

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.