A278458 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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

1,2

Links

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.

Formula

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).

Example

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]  ...

Mathematica

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 *)

Prog

(PARI)
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)))))

Crossrefs

Column k=1 give A000169

Sequence in context: A298663 A325936 A185755 * A309705 A290604 A039796

Adjacent sequences: A278455 A278456 A278457 * A278459 A278460 A278461

Keyword

nonn,tabl

Author

Gheorghe Coserea, Jan 15 2017

Status

approved