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