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

0, 1, 2, 8, 29, 140, 661, 3622, 19993, 120909, 744890, 4887401, 32795272, 230728608, 1661537689, 12426619200, 95087157771, 750968991327, 6062088334528, 50288003979444, 425889463252945, 3694698371069796, 32683415513480237, 295430131502604353, 2719833636188015674, 25536232370225996575

Offset: 0

Author

Gheorghe Coserea, May 28 2017

Keywords

• nonn

Examples

			A(x) = x + 2*x^2 + 8*x^3 + 29*x^4 + 140*x^5 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..301
• Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017.

Crossrefs

Column zero of A287040.

Cf. A262301, A267827, A281270, A287030.

Programs

Mathematica

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}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 13 2018 *)

PARI

A287040_ser(N) = {
  my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
  while(n++,
    F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
    if (F1 == F0, break()); F0 = F1; ); F0;
};
concat(0, Vec(subst(A287040_ser(26), 't, 0)))

PARI

A287045_seq(N) = {
  my(a = vector(N), b=vector(N), t1=0);
  a[1]=1; a[2]=2; a[3]=8; b[1]=0; b[2]=1; b[3]=4;
  for (n=4, N, b[n] = sum(k=1, n-1, a[k]*a[n-k]);
    t1 = 3*a[n-1] + (6*n-10)*a[n-2] - a[n-3];
    a[n] = (t1 + 2*b[n-1] - b[n-2] - b[n-3])/2);
  concat(0,a);
};
A287045_seq(25)
\\ 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

Formula

A(x) = A287040(x;0).

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).

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.