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

0,3

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.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Column zero of A287040.

Cf. A262301, A267827, A281270, A287030.

Sequence in context: A150757 A150758 A185033 * A009419 A000162 A052437

Adjacent sequences: A287042 A287043 A287044 * A287046 A287047 A287048

Keyword

nonn

Author

Gheorghe Coserea, May 28 2017

Status

approved