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The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform' and the Sage implementation.'.

A268442Matrix[dim_] := Module[ {v, r, A},

v = Table[Subscript[x, j], {j, 1, dim}];

r = Table[Subscript[x, j]->1, {j, 1, n}];

A = Table[Table[BellY[n, k, v], {k, 0, dim}], {n, 0, dim}];

Table[Table[MonomialList[Inverse[A][[n, k]]/. r[[1]],

v, Lexicographic] /. r, {k, 1, n}] // Flatten, {n, 1, dim}]];

A268442Matrix[7] // Flatten

Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of first kind (A048994). The column 1 of sublists is A176740 (which missesmissing the leading 1) and A134685 in different order.

The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the Sage code belowlink 'Bell transform' and the link 'BellSage transform'.implementation.

Peter Luschny, <a href="/A268442/b268442.txt">TableFirst of21 nrows, a(n) for n = 0..8286flattened</a>

(Sage)) # see link

def InverseBellPolynomial_List(dim):

def bell_polynomial(n):

X = var(['x'+str(i) for i in (0..dim)])

@cached_function

def T(n, k):

if k==0: return k^n

return sum(binomial(n-1, j-1)*T(n-j, k-1)*X[j-1]

for j in (0..n-k+1)).expand()

return [T(n, k) for k in (0..n)]

A = [[f for f in bell_polynomial(n)] for n in range(dim)]

M = [[0 for k in (0..n)] for n in range(dim)]

for n in range(dim):

M[n][n] = 1/A[n][n]

for k in range(n-1, -1, -1):

M[n][k] = expand(-sum(A[i][k]*M[n][i]

for i in range(n, k, -1))/A[k][k])

return M

import itertools

def Coefficients(M):

def flatten(iter_lst):

L = list(itertools.chain(*iter_lst))

return list(itertools.chain(*L))

def coefficient(p):

c = SR(p).fraction(ZZ).numerator().coefficients()

return [0] if not c else c

A = [[coefficient(p) for p in M[n]] for n in range(len(M))]

return flatten(A)

A268442_matrix = lambda dim: Coefficients(InverseBellPolynomial_List(dim))

A268442_matrix(7)

Peter Luschny, <a href="/A268442/a268442.txt">SageMath implementation</a>

Cf. A036040, A048994, A134685, A176740, A268441.

Peter Luschny, <a href="/A268442/b268442.txt">Table of n, a(n) for n = 0..8286</a>