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

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

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

Examples

			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}
		

Links

• 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

Crossrefs

Row sums give A049425.

Cf. A004747, A049324.

Programs

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 *)

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

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

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

Extensions

New name from Peter Luschny, Jan 16 2016

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