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A287040 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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