A114852 The number of closed lambda calculus terms of size n, where size(lambda x.M)=2+size(M), size(M N)=2+size(M)+size(N), and size(V)=1+i for a variable V bound by the i-th enclosing lambda (corresponding to a binary encoding).

0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 6, 5, 13, 14, 37, 44, 101, 134, 298, 431, 883, 1361, 2736, 4405, 8574, 14334, 27465, 47146, 89270, 156360, 293840, 522913, 978447, 1761907, 3288605, 5977863, 11148652, 20414058, 38071898, 70125402, 130880047

Offset

0,9

Links

Table of n, a(n) for n=0..40.

K. Grygiel and P. Lescanne, Counting terms in the binary lambda calculus, arXiv preprint arXiv:1401.0379 [cs.LO], 2014.

Katarzyna Grygiel and Pierre Lescanne, Counting and Generating Terms in the Binary Lambda Calculus (Extended version), HAL Id: ensl-01229794, 2015.

John Tromp, John's Lambda Calculus and Combinatory Logic Playground

John Tromp, Binary Lambda Calculus and Combinatory Logic

John Tromp, More efficient Haskell program

Formula

a(n) = N(0,n) with

N(k,0) = N(k,1) = 0

N(k,n+2) = (if k>n then 1 else 0) +

N(k+1,n) +

Sum_{i=0..n} N(k,i) * N(k,n-i)

Example

a(8) = 2 because lambda x.lambda y.lambda z.z and lambda x.(x x) are the only two closed lambda terms of size 8.

Maple

A114852T := proc(k, n)
    option remember;
    local a;
    if n = 0 or n = 1 then
        0;
    else
        a := procname(k+1, n-2) ;
        if k > n-2 then
            a := a+1 ;
        fi ;
        a := a+add(procname(k, i)*procname(k, n-i-2), i=0..n-2) ;
    end if;
end proc:
A114852 := proc(n)
    A114852T(0, n) ;
end proc: # R. J. Mathar, Feb 28 2015

Mathematica

S[_, 0] = 0; S[_, 1] = 0; S[m_, n_] := S[m, n] = Boole[m >= n-1] + S[m+1, n-2] + Sum[S[m, k] S[m, n-k-2], {k, 0, n-2}];
a[n_] := S[0, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 23 2017 *)

Prog

(Haskell)
a114852 = closed 0 where
  closed k n = if n<2 then 0 else
    (if n-2<k then 1 else 0) +
    closed (k+1) (n-2) +
    sum [closed k i * closed k (n-2-i) | i <- [0..n-2]]
-- See link for a more efficient version.

Crossrefs

Cf. A114851, A195691.

Sequence in context: A307048 A188652 A333958 * A188048 A191529 A095132

Adjacent sequences: A114849 A114850 A114851 * A114853 A114854 A114855

Keyword

nonn

Author

John Tromp, Feb 20 2006

Status

approved