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

Author

John Tromp, Feb 20 2006

Keywords

• nonn

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A114851, A195691.

Programs

Haskell

a114852 = closed 0 where
  closed k n = if n<2 then 0 else
    (if n-2
				

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 *)

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)