login
Triangle read by rows: T(n,0)=B(n) (the Bell numbers, A000110(n)), T(n,k)=0 for k < 0 or k > n, T(n,k) = T(n-1,k) + T(n-1,k-1) for n >= 1, 0 <= k <= n.
1

%I #10 Feb 14 2022 01:25:06

%S 1,1,1,2,2,1,5,4,3,1,15,9,7,4,1,52,24,16,11,5,1,203,76,40,27,16,6,1,

%T 877,279,116,67,43,22,7,1,4140,1156,395,183,110,65,29,8,1,21147,5296,

%U 1551,578,293,175,94,37,9,1,115975,26443,6847,2129,871,468,269,131,46,10,1

%N Triangle read by rows: T(n,0)=B(n) (the Bell numbers, A000110(n)), T(n,k)=0 for k < 0 or k > n, T(n,k) = T(n-1,k) + T(n-1,k-1) for n >= 1, 0 <= k <= n.

%C Row sums = 1, 2, 5, 13, 36, 109, 369, ...

%C Columns 0,1 and 2 yield A000110, A005001 and A029761, respectively.

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 4, 3, 1;

%e 15, 9, 7, 4, 1;

%e 52, 24, 16, 11, 5, 1;

%e 203, 76, 40, 27, 16, 6, 1;

%e ...

%e (4,3) = 16 = 7 + 9 = (3,3) + (3,2).

%p with(combinat): T:=proc(n,k) if k=0 then bell(n) elif k<0 or k>n then 0 else T(n-1,k)+T(n-1,k-1) fi end: for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%Y Cf. A000110, A005001, A029761.

%K nonn,tabl

%O 0,4

%A _Gary W. Adamson_, Nov 22 2006

%E Edited by _N. J. A. Sloane_, Nov 29 2006