A259356 - OEIS

%I #16 Aug 16 2015 16:15:01

%S 0,0,1,0,1,2,0,2,9,3,0,5,38,35,4,0,14,181,284,95,5,0,42,938,2225,1320,

%T 210,6,0,132,5210,17816,15810,4596,406,7

%N Triangle T(n,k) read by rows: T(n,k) is the number of closed lambda-terms of size n with size 0 for the variables and k abstractions.

%F T(n,k) = T(n,k,0) where T(n,k,b) where n is size, k is number of abstractions, and b is number of free variables, T(0,0,b) = b, and T(n,k,b) = T(n-1,k-1,b+1) + Sum_{i=0..n-1} Sum_{j=0..k} T(i,j,b) * T(n-1-i,k-j,b).

%F T(n+1,1) = A000108(n).

%e In table format, the first few rows:

%e {0},

%e {0,1},

%e {0,1,2},

%e {0,2,9,3},

%e {0,5,38,35,4},

%e ...

%e For n=3,k=2 we have the number of closed lambda terms of size three with exactly two abstractions, T(3,2,0) = 9:

%e \x.\y.x x

%e \x.\y.x y

%e \x.\y.y x

%e \x.\y.y y

%e (\x.x) (\y.y)

%e \x.(\y.y) x

%e \x.(\y.x) x

%e \x.x (\y.y)

%e \x.x (\y.x)

%Y Cf. A220894 (row sums), A000108.

%K nonn,tabl,more

%O 0,6

%A _John Bodeen_, Jun 24 2015