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

Author

Gheorghe Coserea, Jan 15 2017

Keywords

• nonn
• tabl

Examples

			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]  ...
		

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.

Crossrefs

Column k=1 give A000169

Programs

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 *)

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)))))

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).