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

0,10

Links

Table of n, a(n) for n=0..74.

Peter Luschny, The Bell transform

Example

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]

Prog

(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))

Crossrefs

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

Cf. A256894.

Sequence in context: A373361 A328392 A051794 * A110969 A362681 A320077

Adjacent sequences: A333302 A333303 A333304 * A333306 A333307 A333308

Keyword

nonn,tabf

Author

Peter Luschny, May 19 2020

Status

approved