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%I #38 Dec 13 2018 08:07:33
%S 0,1,1,1,1,2,5,3,2,8,17,22,10,5,29,91,106,94,35,14,140,431,701,582,
%T 396,126,42,661,2501,4067,4544,2980,1654,462,132,3622,14025,27394,
%U 31032,26680,14598,6868,1716,429,19993,87947,177018,236940,208780,146862,69356,28396,6435,1430,120909,550811,1245517,1727148,1776310,1291654,772422,322204,117016,24310,4862
%N Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%H Gheorghe Coserea, <a href="/A287040/b287040.txt">Rows n=0..200, flattened</a>
%H Pierre Lescanne, <a href="https://arxiv.org/abs/1702.03085">Quantitative aspects of linear and affine closed lambda terms</a>, arXiv:1702.03085 [cs.DM], 2017.
%F y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = t + x*y^2 + x*deriv(y,t) + x*y, with y(0;t)=t, where P_n(t) = Sum_{k=0..n+1} T(n,k)*t^k.
%F A000108(n)=T(n,n+1), A001700(n)=T(n+1,n+1).
%e A(x;t) = t + (1 + t + t^2)*x + (2 + 5*t + 3*t^2 + 2*t^3)*x^2 + (8 + 17*t + 22*t^2 + 10*t^3 + 5*t^4)*x^3 + ...
%e Triangle starts:
%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e [0] 0, 1;
%e [1] 1, 1, 1;
%e [2] 2, 5, 3, 2;
%e [3] 8, 17, 22, 10, 5;
%e [4] 29, 91, 106, 94, 35, 14;
%e [5] 140, 431, 701, 582, 396, 126, 42;
%e [6] 661, 2501, 4067, 4544, 2980, 1654, 462, 132;
%e [7] 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429;
%e [8] 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430;
%e [9] ...
%t nmax = 10; y[0, t_] := t; y[_, _] = 0;
%t Do[y[x_, t_] = Series[t + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, nmax}, {t, 0, nmax}] // Normal, {n, 0, nmax}];
%t CoefficientList[#, t]& /@ CoefficientList[y[x, t]+O[x]^nmax, x] // Flatten (* _Jean-François Alcover_, Dec 13 2018 *)
%o (PARI)
%o A287040_ser(N) = {
%o my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
%o while(n++,
%o F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
%o if (F1 == F0, break()); F0 = F1; ); F0;
%o };
%o concat(apply(p->Vecrev(p), Vec(A287040_ser(10))))
%o \\ test: y=A287040_ser(50); y == t + x*y^2 + x*deriv(y, t) + x*y
%Y Cf. A262301, A267827, A281270, A287030, A287045 (column 0).
%K nonn,tabl
%O 0,6
%A _Gheorghe Coserea_, May 23 2017
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