

A275057


Numbers of closed lambda terms of natural size n.


2



0, 0, 1, 1, 3, 6, 17, 41, 116, 313, 895, 2550, 7450, 21881, 65168, 195370, 591007, 1798718, 5510023, 16966529, 52506837, 163200904, 509323732, 1595311747, 5013746254, 15805787496, 49969942138, 158396065350, 503317495573, 1602973785463, 5116010587910, 16360492172347
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OFFSET

0,5


COMMENTS

Natural size measure lambda terms as follows: all symbols are assigned size 1, namely applications, abstractions, successor symbols in de Bruijn indices and 0 symbol in de Bruijn indices (i.e., a de Bruijn index n is assigned size n+1).
Here we count the closed terms of natural size n, where "closed" means that there is no free index (no free bound variable).


LINKS

Pierre Lescanne, Table of n, a(n) for n = 0..299
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, SOFSEM 2016: 183194.
Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simplytyped lambdaterms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 20162017.


FORMULA

L(0,m) = 0.
L(n+1,m) = (Sum_{k=0..n} L(k,m)*L(nk,m)) + L(n,m+1) + [m >= n+1], where [p(n,m)] = 1 if p(n,m) is true and [p(n,m)] = 0 if p(n,m) is false then one considers the sequence (L(n,0)).


MATHEMATICA

L[0, _] = 0; L[n_, m_] := L[n, m] = Sum[L[k, m]*L[nk1, m], {k, 0, n1}] + L[n1, m+1] + Boole[m >= n];
a[n_] := L[n, 0];
Table[a[n], {n, 0, 31}] (* JeanFrançois Alcover, May 23 2017 *)


CROSSREFS

Cf. A105633, A272794.
Sequence in context: A319789 A007718 A297972 * A320807 A089264 A121399
Adjacent sequences: A275054 A275055 A275056 * A275058 A275059 A275060


KEYWORD

nonn


AUTHOR

Pierre Lescanne, Jul 14 2016


STATUS

approved



