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A287030 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A287030 #25 Oct 25 2018 08:14:50
%S A287030 0,0,1,1,1,2,1,1,3,5,3,9,17,6,2,30,41,26,10,81,131,111,30,5,242,491,
%T A287030 357,134,35,838,1625,1274,652,140,14,2799,5497,5202,2556,676,126,9365,
%U A287030 20581,19827,10200,3610,630,42,33616,76561,74797,44880,16390,3334,462,122937,282591,301188,190278,72490,19218,2772,132,449698,1089375,1219920,788654,341770,97890,16108,1716,1696724,4285737,4893603,3398950,1578577,474838,99386,12012,429
%N A287030 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A287030 Row n contains floor((n+3)/2) terms.
%H A287030 Gheorghe Coserea, <a href="/A287030/b287030.txt">Rows n=0..200, flattened</a>
%H A287030 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 A287030 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = t*x + x*y^2 + x*deriv(y,t) + x*y, with y(0;t)=0, where P_n(t) = Sum_{k=0..floor((n+1)/2)} T(n,k)*t^k.
%F A287030 A281270(n)=T(n,0), A000108(n)=T(2*n+1,n+1), A001700(n-1)=T(2*n,n).
%e A287030 A(x;t) = t*x + (1 + t)*x^2 + (2 + t + t^2)*x^3 + (3 + 5*t + 3*t^2)*x^4 + (9 + 17*t + 6*t^2 + 2*t^3)*x^5 + ...
%e A287030 Triangle starts:
%e A287030 n\k   [0]     [1]      [2]      [3]     [4]     [5]    [6]    [7]
%e A287030 [0]   0;
%e A287030 [1]   0,      1;
%e A287030 [2]   1,      1;
%e A287030 [3]   2,      1,       1;
%e A287030 [4]   3,      5,       3;
%e A287030 [5]   9,      17,      6,       2;
%e A287030 [6]   30,     41,      26,      10;
%e A287030 [7]   81,     131,     111,     30,     5;
%e A287030 [8]   242,    491,     357,     134,    35;
%e A287030 [9]   838,    1625,    1274,    652,    140,    14;
%e A287030 [10]  2799,   5497,    5202,    2556,   676,    126;
%e A287030 [11]  9365,   20581,   19827,   10200,  3610,   630,   42;
%e A287030 [12]  33616,  76561,   74797,   44880,  16390,  3334,  462;
%e A287030 [13]  122937, 282591,  301188,  190278, 72490,  19218, 2772,  132;
%e A287030 [14]  449698, 1089375, 1219920, 788654, 341770, 97890, 16108, 1716;
%e A287030 [15]  ...
%t A287030 max = 15; y[_, _] = 0;
%t A287030 Do[y[x_, t_] = Series[t x + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, max}, {t, 0, max}] // Normal, max];
%t A287030 CoefficientList[#, t]& /@ CoefficientList[y[x, t], x] /. {} -> {0} // Flatten (* _Jean-François Alcover_, Oct 25 2018 *)
%o A287030 (PARI)
%o A287030 A287030_ser(N) = {
%o A287030   my(x='x+O('x^N), F0=x, t='t, F1=0, n=1);
%o A287030   while(n++,
%o A287030     F1 = t*x + x*F0^2 + x*deriv(F0,t) + x*F0;
%o A287030     if (F1 == F0, break()); F0 = F1;); F0;
%o A287030 };
%o A287030 concat(0, concat(apply(p->Vecrev(p), Vec(A287030_ser(16)))))
%o A287030 \\ test: y=A287030_ser(100); y == t*x + x*y^2 + x*deriv(y,t) + x*y
%Y A287030 Cf. A262301, A267827, A281270, A287040, A287045.
%K A287030 nonn,tabf
%O A287030 0,6
%A A287030 _Gheorghe Coserea_, May 23 2017