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A287045 a(n) is the number of size n affine closed terms of variable size 0.

%I A287045 #26 Dec 13 2018 08:07:26
%S A287045 0,1,2,8,29,140,661,3622,19993,120909,744890,4887401,32795272,
%T A287045 230728608,1661537689,12426619200,95087157771,750968991327,
%U A287045 6062088334528,50288003979444,425889463252945,3694698371069796,32683415513480237,295430131502604353,2719833636188015674,25536232370225996575
%N A287045 a(n) is the number of size n affine closed terms of variable size 0.
%H A287045 Gheorghe Coserea, <a href="/A287045/b287045.txt">Table of n, a(n) for n = 0..301</a>
%H A287045 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 A287045 A(x) = A287040(x;0).
%F A287045 a(n) = (3*a(n-1) + (6*n-10)*a(n-2) - a(n-3) + 2*b(n-1) - b(n-2) - b(n-3))/2, where b(n) = Sum_{k=1..n-1} a(k)*a(n-k).
%F A287045 0 = 6*x^3*deriv(y,x) - x*(x-1)*(x+2)*y^2 - (x^3-2*x^2-3*x+2)*y + x^2 + 2*x, where y(x) is the g.f.
%e A287045 A(x) = x + 2*x^2 + 8*x^3 + 29*x^4 + 140*x^5 + ...
%t A287045 a[n_] := a[n] = If[n<3, n, (3a[n-1] + (6n-10) a[n-2] - a[n-3] + 2b[n-1] - b[n-2] - b[n-3])/2]; b[n_] := Sum[a[k] a[n-k], {k, 1, n-1}];
%t A287045 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 13 2018 *)
%o A287045 (PARI)
%o A287045 A287040_ser(N) = {
%o A287045   my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
%o A287045   while(n++,
%o A287045     F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
%o A287045     if (F1 == F0, break()); F0 = F1; ); F0;
%o A287045 };
%o A287045 concat(0, Vec(subst(A287040_ser(26), 't, 0)))
%o A287045 (PARI)
%o A287045 A287045_seq(N) = {
%o A287045   my(a = vector(N), b=vector(N), t1=0);
%o A287045   a[1]=1; a[2]=2; a[3]=8; b[1]=0; b[2]=1; b[3]=4;
%o A287045   for (n=4, N, b[n] = sum(k=1, n-1, a[k]*a[n-k]);
%o A287045     t1 = 3*a[n-1] + (6*n-10)*a[n-2] - a[n-3];
%o A287045     a[n] = (t1 + 2*b[n-1] - b[n-2] - b[n-3])/2);
%o A287045   concat(0,a);
%o A287045 };
%o A287045 A287045_seq(25)
%o A287045 \\ test: y=Ser(A287045_seq(200)); 0 == 6*x^3*y' - x*(x-1)*(x+2)*y^2 - (x^3-2*x^2-3*x+2)*y + x^2 + 2*x
%Y A287045 Column zero of A287040.
%Y A287045 Cf. A262301, A267827, A281270, A287030.
%K A287045 nonn
%O A287045 0,3
%A A287045 _Gheorghe Coserea_, May 28 2017