1,2

Previous name was: A triangle of numbers related to triangle A049326.

a(n,1) = A008279(4,n-1). a(n,m) =: S1(-4; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A011801(n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Peter Luschny, The Bell transform

a(n, m) = n!*A049326(n, m)/(m!*5^(n-m));

a(n, m) = (5*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;

a(n, m) = 0, n<m; a(n, 0) := 0; a(1, 1) = 1.

E.g.f. for m-th column: (((-1+(1+x)^5)/5)^m)/m!.

E.g., row polynomial E(3,x) = 12*x + 12*x^2 + x^3.

Triangle starts:

1;

4, 1;

12, 12, 1;

24, 96, 24, 1;

24, 600, 360, 40, 1;

rows = 10;

a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];

Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

(SageMath) # uses[bell_matrix from A264428]

# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.

bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

Cf. A049326.

Row sums give A049427.

Sequence in context: A227338 A125105 A144878 * A157394 A338864 A078219

Adjacent sequences: A049421 A049422 A049423 * A049425 A049426 A049427

easy,nonn,tabl

New name from Peter Luschny, Jan 16 2016

approved