A135501 Number of closed lambda-terms of size n with size 1 for the variables.

0, 0, 1, 2, 4, 13, 42, 139, 506, 1915, 7558, 31092, 132170, 580466, 2624545, 12190623, 58083923, 283346273, 1413449148, 7200961616, 37425264180, 198239674888, 1069228024931, 5867587726222, 32736878114805, 185570805235978

Offset: 0

Author

Christophe Raffalli (christophe.raffalli(AT)univ-savoie.fr), Feb 09 2008

Keywords

• nonn

Comments

A lambda-term is a term which is either a variable "x" (of size 1), an application of two lambda-terms (of size 1 + the sum of the sizes of the two subterms), or a lambda binding a new variable in a term (of size 1 + the size of the subterm).

The size of a lambda-term t can be equivalently defined as the number of ways of selecting a subterm of t (distinguishing between different occurrences of the same subterm). For example, the 2 closed terms with three subterms are \x.\y.x (subterms \x.\y.x, \y.x, and x) and \x.\y.y (subterms \x.\y.y, \y.y, and y), while the 4 with four subterms are \x.\y.\z.x, \x.\y.\z.y, \x.\y.\z.z, and \x.x(x) (subterms \x.x(x), x(x), x[1], and x[2], distinguishing between the two occurrences of the variable x). - Noam Zeilberger, Nov 10 2018

Links

• Noam Zeilberger, Table of n, a(n) for n = 0..300 (terms 2-63 from Christophe Raffalli)
• Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, On the number of unary-binary tree-like structures with restrictions on the unary height, arXiv:1510.01167 [math.CO], 2015, see p. 4.
• Olivier Bodini, Paul Tarau, On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms, arXiv:1709.04302 [cs.PL], 2017.
• Katarzyna Grygiel and Pierre Lescanne, Counting and generating lambda-terms, arXiv preprint arXiv:1210.2610 [cs.LO], 2012-2013.
• Pierre Lescanne, On counting untyped lambda terms, arXiv:1107.1327 [cs.LO], 2011-2012.
• Paul Tarau, On Type-directed Generation of Lambda Terms, preprint, 2015.
• Paul Tarau, On logic programming representations of lambda terms: de Bruijn indices, compression, type inference, combinatorial generation, normalization, 2015.
• P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.
• Noam Zeilberger, Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014.

Crossrefs

Cf. A220894, A062980.

Programs

Maple

T := proc(n, k) option remember; `if`(n <= 0, 0, `if`(n = 1, k,
T(n-1, k+1) + add(T(n-1-j, k)*T(j, k), j = 0..n-2))) end:
a := n -> T(n, 0): seq(a(n), n=0..25); # Peter Luschny, Nov 10 2018

Mathematica

T[0, ] = 0; T[1, m] := m;
T[n_, m_] := T[n, m] = T[n-1, m+1] + Sum[T[n-1-k, m] T[k, m], {k, 0, n-2}];
a[n_] := T[n, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 06 2018, after Pierre Lescanne *)

Formula

f(n,0) where f(1,1) = 1; f(0,k) = 0; f(n,k) = 0 if k>2n-1; f(n,k) = f(n-1,k) + f(n-1,k+1) + sum_{p=1 to n-2} sum_{c=0 to k} sum_{l=0 to k - c} [C^c_k C^l_(k-c) f(p,l+c) f(n-p-1,k-l)], where C^p_n are binomial coefficients (the last term is for the application where "c" is the number of common variables in both subterms). f(n,k) can be computed only using f(n',k') with n' < n and k' <= k + n - n'. f(n,k) is the number of lambda terms of size n (with size 1 for the variables) having exactly k free variables.

T(n,0) where T(0,m) = 0; T(1,m) = m; T(n+1,m) = T(n,m+1) + sum{k=0 to n-11} [T(n-k,m) T(k,m)]. T(n,m) is the number of lambda terms of size n (with size 1 for the variables) having at most m free variables. [Pierre Lescanne, Nov 18 2012]

The ordinary generating function l(z) can be computed using a catalytic variable as l(z) = L(z,0), where L(z,f) satisfies the functional equation L(z,f) = f*z + z*L(z,f)^2 + z*L(z,f+1). See equation (2) of [Bodini et al., 2015]. - Noam Zeilberger, Nov 10 2018

Extensions

Two terms prepended by Noam Zeilberger, Nov 10 2018