A333305 Irregular array read by rows, a refinement of A256894.

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

Offset: 0

Author

Peter Luschny, May 19 2020

Keywords

• nonn
• tabf

Examples

			Irregular table (the refinement is indicated by round brackets) starts:
[0] [1]
[1] [1, 1]
[2] [1, (1, 1), 1]
[3] [1, (1, 2, 1), (3, 1), 1]
[4] [1, (1, 3, 3, 1), (4, 3, 5, 1), (6, 1), 1]
[5] [1, (1, 4, 6, 4, 1), (5, 10, 9, 8, 7, 1), (10, 15, 9, 1), (10, 1), 1]
[6] [1, (1, 5, 10, 10, 5, 1), (6, 15, 14, 10, 35, 16, 15, 9, 1), (15, 60, 19, 15,
     33, 12, 1), (20, 45, 14, 1), (15, 1), 1]
		

Links

• Peter Luschny, The Bell transform

Crossrefs

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

Cf. A256894.

Programs

SageMath

def BellBlocks(n):
    R = InfinitePolynomialRing(ZZ, 'v') # Thanks to F. Chapoton.
    V = R.gen()
    @cached_function
    def T(n, k):
        if k == 0: return V[0]^n
        return sum(binomial(n-1, j-1)*V[j]*T(n-j, k-1) for j in (0..n-k+1))
    P = (T(n, k) for k in (0..n))
    return flatten([p.coefficients() for p in P])
for n in (0..8): print(BellBlocks(n))