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A114852
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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).
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6
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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
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OFFSET
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0,9
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REFERENCES
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Katarzyna Grygiel, Pierre Lescanne. Counting and Generating Terms in the Binary Lambda Calculus (Extended version). 2015. <ensl-01229794>
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LINKS
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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
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FORMULA
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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)
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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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.
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CROSSREFS
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Cf. A114851, A195691.
Sequence in context: A156993 A030770 A188652 * A188048 A191529 A095132
Adjacent sequences: A114849 A114850 A114851 * A114853 A114854 A114855
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KEYWORD
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nonn
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AUTHOR
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John Tromp, Feb 20 2006
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STATUS
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approved
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