A262301 Number of normal linear lambda terms of size n with no free variables.

1, 3, 26, 367, 7142, 176766, 5304356, 186954535, 7566084686, 345664350778, 17592776858796, 986961816330662, 60502424162842876, 4023421969420255644, 288464963899330354104, 22180309834307193611287, 1820641848410408158704734, 158897008602951290424279330

Offset: 1

Author

N. J. A. Sloane, Sep 30 2015

Keywords

• nonn

Examples

			A(x) = x + 3*x^2 + 26*x^3 + 367*x^4 + 7142*x^5 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..100
• Paul Tarau, Valeria de Paiva, Deriving Theorems in Implicational Linear Logic, Declaratively, arXiv:2009.10241 [cs.LO], 2020. See also Github, (2020).
• Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
• Wikipedia, Lambda calculus

Crossrefs

Column 0 of A318110.

Cf. A062980, A267827.

Programs

Mathematica

terms = 18; F[, ] = 0;
Do[F[x_, t_] = Series[x t/(1-F[x, t]) + D[F[x, t], t], {x, 0, terms}, {t, 0, terms}] // Normal, {2 terms}];
CoefficientList[F[x, 0], x][[2 ;; terms+1]] (* Jean-François Alcover, Sep 02 2018, after Gheorghe Coserea *)

PARI

F(N) = {
  my(x='x+O('x^N), t='t, F0=x, F1=0, n=1);
  while(n++,
    F1 = x*t/(1-F0) + deriv(F0,t);
    if (F1 == F0, break()); F0 = F1;);
  F0;
};
seq(N) = Vec(subst(F(N+1), 't, 0));
seq(18) \\ Gheorghe Coserea, Apr 01 2017

Formula

A(x) = F(x,0), where A(x) = Sum_{n>=1} a(n)*x^n and F(x,t) satisfies F = x*t/(1-F) + deriv(F,t), with F(0,t)=0, deriv(F,x)(0,t)=1+t. - Gheorghe Coserea, Apr 01 2017

Extensions

More terms from Gheorghe Coserea, Apr 01 2017