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

%I A278458 #39 Sep 28 2019 12:30:42
%S A278458 1,2,2,9,15,8,64,156,144,52,625,2050,2675,1730,472,7776,32430,55000,
%T A278458 50310,25108,5504,117649,599319,1258775,1484245,1052184,428036,78416,
%U A278458 2097152,12669496,31902416,46103680,42064736,24421096,8389552,1320064,43046721,301574340,888996066,1524644856,1698413409,1269814980,625219644,185935104,25637824
%N A278458 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%H A278458 Gheorghe Coserea, <a href="/A278458/b278458.txt">Rows n = 1..101, flattened</a>.
%H A278458 F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%F A278458 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.
%F A278458 A006351(n) = P_n(0), A005172(n) = P_n(1), A231691(n) = P_n(2).
%e A278458 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! + ...
%e A278458 Triangle starts:
%e A278458 n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
%e A278458 [1]  1;
%e A278458 [2]  2,       2;
%e A278458 [3]  9,       15,      8;
%e A278458 [4]  64,      156,     144,     52;
%e A278458 [5]  625,     2050,    2675,    1730,    472;
%e A278458 [6]  7776,    32430,   55000,   50310,   25108,   5504;
%e A278458 [7]  117649,  599319,  1258775, 1484245, 1052184, 428036,  78416;
%e A278458 [8]  ...
%t A278458 m = 10;
%t A278458 (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 *)
%o A278458 (PARI)
%o A278458 N=10; x = 'x + O('x^N); t='t;
%o A278458 concat(apply(p->Vec(p), Vec(serlaplace(serreverse(log(x + exp(t*log(1+x))) - (t-1)*log(1+x) - x)))))
%Y A278458 Column k=1 give A000169
%K A278458 nonn,tabl
%O A278458 1,2
%A A278458 _Gheorghe Coserea_, Jan 15 2017