A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).

1, 2, 1, 2, 6, 1, 0, 20, 12, 1, 0, 40, 80, 20, 1, 0, 40, 360, 220, 30, 1, 0, 0, 1120, 1680, 490, 42, 1, 0, 0, 2240, 9520, 5600, 952, 56, 1, 0, 0, 2240, 40320, 48720, 15120, 1680, 72, 1, 0, 0, 0, 123200, 332640, 184800, 35280, 2760, 90, 1, 0, 0, 0, 246400, 1786400

Offset

1,2

Comments

Previous name was: A triangle of numbers related to triangle A049324.

a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; 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))*A004747(n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), 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

Links

Table of n, a(n) for n=1..60.

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

W. Lang, First 10 rows of the array and more. [From Wolfdieter Lang, Oct 17 2008]

Peter Luschny, The Bell transform

Formula

a(n, m) = n!*A049324(n, m)/(m!*3^(n-m));

a(n, m) = (3*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: ((x+x^2+(x^3)/3)^m)/m!.

a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - Werner Schulte, Feb 20 2020

Example

E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
Triangle starts:
{1}
{2,  1}
{2,  6,  1}
{0, 20, 12, 1}

Mathematica

rows = 11;
a[n_, m_] := BellY[n, m, Table[k! Binomial[2, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Prog

(Sage) # 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

Crossrefs

Row sums give A049425.

Cf. A004747, A049324.

Sequence in context: A144824 A364513 A144358 * A159885 A178803 A292901

Adjacent sequences: A049401 A049402 A049403 * A049405 A049406 A049407

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

New name from Peter Luschny, Jan 16 2016

Status

approved