A231690 -id:A231690 - 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]

A231691 Cardinalities of the symmetric operad of dotted red and white trees.

Original entry on oeis.org

1, 6, 74, 1476, 41032, 1464672, 63865328, 3290120832, 195537380704, 13169097667584, 991181618539136, 82450282595311104, 7511417235983147008, 743790032122343820288, 79541198937597284060672, 9136079502141558495310848, 1121720442822518015112749056, 146607501639123412303738884096, 20322509742114322789584125210624, 2978025324234142178848508363882496

Offset: 1

N. J. A. Sloane, Nov 14 2013

nonn

			A(x) = x + 6*x^2/2! + 74*x^3/3! + 1476*x^4/4! + 41032*x^5/5! + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..300
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Cf. A052878, A231690.

S:= series(RootOf(y=-x-ln((1+x)/(1+3*x+x^2)),x),y,21):
seq(coeff(S,y,n)*n!,n=1..21); # Robert Israel, Sep 27 2018

terms = 20; (CoefficientList[InverseSeries[Log[x^2 + 3x + 1] - Log[1+x] - x + O[x]^(terms+1)], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 16 2018, after Gheorghe Coserea *)

N=21; x = 'x + O('x^N); Vec(serlaplace(serreverse(log(x^2+3*x+1) - log(1+x) - x))) \\ Gheorghe Coserea, Jan 18 2017

E.g.f. A(x) satisfies -A(x) - g(-A(x)) = x where g is the E.g.f. of A052878. - Gheorghe Coserea, Jan 18 2017, edited by Robert Israel, Sep 27 2018

a(n) ~ sqrt((5 + 7*s + 3*s^2) / (7 + 13*s + 5*s^2)) * n^(n-1) / ((log((1+3*s+s^2)/(1+s))-s)^(n - 1/2) * exp(n)), where s = A060006 - 1 = -1 + (27/2 - 3*sqrt(69)/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3). - Vaclav Kotesovec, Apr 21 2020

Offset changed and more terms from Gheorghe Coserea, Jan 15 2017

A278457 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 2, 7, 11, 6, 30, 65, 59, 22, 143, 397, 492, 318, 90, 728, 2471, 3857, 3430, 1728, 394, 3876, 15572, 29255, 32728, 22886, 9461, 1806, 21318, 99009, 217323, 291456, 257001, 148626, 52133, 8558, 120175, 633765, 1591231, 2481597, 2622445, 1918665, 947740, 288812, 41586, 690690, 4078360, 11527318, 20467755, 25114375, 22043890, 13821764, 5964728, 1607198, 206098

Offset: 1

Gheorghe Coserea, Jan 15 2017

			A(x;t) = x + (2*t+2)*x^2 + (7*t^2+11*t+6)*x^3 + (30*t^3+65*t^2+59*t+22)*x^4 + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
[1]  1;
[2]  2,       2;
[3]  7,       11,      6;
[4]  30,      65,      59,      22;
[5]  143,     397,     492,     318,     90;
[6]  728,     2471,    3857,    3430,    1728,    394;
[7]  3876,    15572,   29255,   32728,   22886,   9461,    1806;
[8]  21318,   99009,   217323,  291456,  257001,  148626,  52133,   8558;
[9]  ...

Gheorghe Coserea, Rows n = 1..101, flattened
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Column k=1 give A006013.

Reverse[CoefficientList[#, t]]& /@ CoefficientList[InverseSeries[x ((t - 1) x^3 + (t^2 - 2t - 1) x^2 - (2t - 1) x + 1)/((1 - t) x^3 + (3 - t) x^2 + 3x + 1) + O[x]^11], x] // Flatten (* Jean-François Alcover, Sep 28 2019 *)

N=11; x ='x + O('x^N);
concat(apply(p->Vec(p), Vec(serreverse(Ser(x*((t-1)*x^3 + (t^2-2*t-1)*x^2 - (2*t-1)*x+1)/((1-t)*x^3 + (3-t)*x^2 + 3*x + 1), 'x)))))

y(x) = Sum {n>=1} P_n(t)*x^n satisfies x = y*((t-1)*y^3 + (t^2-2*t-1)*y^2 - (2*t-1)*y + 1)/((1-t)*y^3 + (3-t)*y^2 + 3*y + 1), with y(0)=0, y'(0)=1, where P_n(t) is the degree n-1 polynomial associated with row n of the triangle in order of decreasing powers of t.

P_n(0) = A006318(n-1), P_n(1) = A156017(n-1), P_n(2) = A231690(n).

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).

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A231691 Cardinalities of the symmetric operad of dotted red and white trees.

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A278457 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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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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Author

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Comments

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Crossrefs

Programs

Mathematica

A231691 Cardinalities of the symmetric operad of dotted red and white trees.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A278457 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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).