A231691 -id:A231691 - cp's OEIS frontend

A231690 Cardinalities of the sub-operad FF_6 of the operad MFF.

1, 6, 56, 640, 8158, 111258, 1588544, 23446248, 354855218, 5477342222, 85893429256, 1364577254040, 21916000458014, 355251287893170, 5804407209709312, 95493879511032240, 1580592247322440642, 26301843121772151254, 439764358275666481496, 7384252698468635017936, 124469446338979722639294

Offset: 1

N. J. A. Sloane, Nov 14 2013

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..301
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. A213667, A231691.

InverseSeries[x(1 - 3x - x^2 + x^3)/(1 + 3x + x^2 - x^3) + O[x]^22] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Oct 24 2018, from PARI *)

N=22; x='x+O('x^N);
Vec(serreverse(Ser(x*(1-3*x-x^2+x^3)/(1+3*x+x^2-x^3))))  \\ Gheorghe Coserea, Jan 13 2017

A(x) = serreverse(A213667(-x))(-x). - Gheorghe Coserea, Jan 13 2017

Offset changed and more terms from Gheorghe Coserea, Jan 13 2017

A278458 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, 9, 15, 8, 64, 156, 144, 52, 625, 2050, 2675, 1730, 472, 7776, 32430, 55000, 50310, 25108, 5504, 117649, 599319, 1258775, 1484245, 1052184, 428036, 78416, 2097152, 12669496, 31902416, 46103680, 42064736, 24421096, 8389552, 1320064, 43046721, 301574340, 888996066, 1524644856, 1698413409, 1269814980, 625219644, 185935104, 25637824

Offset: 1

Gheorghe Coserea, Jan 15 2017

			A(x;t) = x + (2*t+2)*x^2/2! + (9*t^2+15*t+8)*x^3/3! + (64*t^3+156*t^2+144*t+52)*x^4/4! + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
[1]  1;
[2]  2,       2;
[3]  9,       15,      8;
[4]  64,      156,     144,     52;
[5]  625,     2050,    2675,    1730,    472;
[6]  7776,    32430,   55000,   50310,   25108,   5504;
[7]  117649,  599319,  1258775, 1484245, 1052184, 428036,  78416;
[8]  ...

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 A000169

m = 10;
(Reverse[CoefficientList[#, t]]& /@ CoefficientList[InverseSeries[Log[x + Exp[t Log[1+x]]] - (t-1) Log[1+x] - x + O[x]^m], x]) Range[0, m-1]! // Rest // Flatten (* Jean-François Alcover, Sep 28 2019 *)

N=10; x = 'x + O('x^N); t='t;
concat(apply(p->Vec(p), Vec(serlaplace(serreverse(log(x + exp(t*log(1+x))) - (t-1)*log(1+x) - x)))))

y(x;t) = Sum {n>=1} P_n(t)*x^n/n! satisfies x = log(y + exp(t*log(1+y))) - (t-1)*log(1+y) - y.

A006351(n) = P_n(0), A005172(n) = P_n(1), A231691(n) = P_n(2).

cp's OEIS Frontend

A231690 Cardinalities of the sub-operad FF_6 of the operad MFF.

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