%I #17 May 26 2020 16:50:43

%S 1,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,3,3,1,4,3,5,1,6,1,1,1,1,4,6,4,1,5,10,

%T 9,8,7,1,10,15,9,1,10,1,1,1,1,5,10,10,5,1,6,15,14,10,35,16,15,9,1,15,

%U 60,19,15,33,12,1,20,45,14,1,15,1,1

%N Irregular array read by rows, a refinement of A256894.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>

%e Irregular table (the refinement is indicated by round brackets) starts:

%e [0] [1]

%e [1] [1, 1]

%e [2] [1, (1, 1), 1]

%e [3] [1, (1, 2, 1), (3, 1), 1]

%e [4] [1, (1, 3, 3, 1), (4, 3, 5, 1), (6, 1), 1]

%e [5] [1, (1, 4, 6, 4, 1), (5, 10, 9, 8, 7, 1), (10, 15, 9, 1), (10, 1), 1]

%e [6] [1, (1, 5, 10, 10, 5, 1), (6, 15, 14, 10, 35, 16, 15, 9, 1), (15, 60, 19, 15,

%e 33, 12, 1), (20, 45, 14, 1), (15, 1), 1]

%o (SageMath)

%o def BellBlocks(n):

%o R = InfinitePolynomialRing(ZZ, 'v') # Thanks to _F. Chapoton_.

%o V = R.gen()

%o @cached_function

%o def T(n, k):

%o if k == 0: return V[0]^n

%o return sum(binomial(n-1, j-1)*V[j]*T(n-j, k-1) for j in (0..n-k+1))

%o P = (T(n, k) for k in (0..n))

%o return flatten([p.coefficients() for p in P])

%o for n in (0..8): print(BellBlocks(n))

%Y Cf. A000070 (length of rows), A102356 (max in rows), A186021 (sum of rows).

%Y Cf. A256894.

%K nonn,tabf

%O 0,10

%A _Peter Luschny_, May 19 2020