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A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0). 7
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 (list; table; graph; refs; listen; history; text; internal format)
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!.

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}

PROG

(Sage)

# The function bell_matrix is defined in 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: A006602 A144824 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

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Last modified February 25 13:30 EST 2018. Contains 299654 sequences. (Running on oeis4.)