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