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A287045 a(n) is the number of size n affine closed terms of variable size 0. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 14 15:15 EST 2019. Contains 329126 sequences. (Running on oeis4.)