|
|
A049424
|
|
Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).
|
|
3
|
|
|
1, 4, 1, 12, 12, 1, 24, 96, 24, 1, 24, 600, 360, 40, 1, 0, 3024, 4200, 960, 60, 1, 0, 12096, 40824, 17640, 2100, 84, 1, 0, 36288, 338688, 270144, 55440, 4032, 112, 1, 0, 72576, 2407104, 3580416, 1212624, 144144, 7056, 144, 1, 0, 72576, 14515200, 41791680
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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).
|
|
LINKS
|
|
|
FORMULA
|
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!.
|
|
EXAMPLE
|
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;
|
|
MATHEMATICA
|
rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|