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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). 8
 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 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, 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

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Last modified January 24 13:11 EST 2022. Contains 350538 sequences. (Running on oeis4.)