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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). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

K. Grygiel, P. Lescanne, Counting terms in the binary lambda calculus, arXiv preprint arXiv:1401.0379, 2014

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

PROG

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: A156993 A030770 A188652 * A188048 A191529 A095132

Adjacent sequences:  A114849 A114850 A114851 * A114853 A114854 A114855

KEYWORD

nonn

AUTHOR

John Tromp, Feb 20 2006

STATUS

approved

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Last modified August 28 18:07 EDT 2015. Contains 261154 sequences.