OFFSET
1,2
COMMENTS
This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)
EXAMPLE
[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From Philippe Deléham, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
1;
0, 1;
0, 3, 1;
0, 9, 9, 1;
0, 27, 63, 18, 1;
0, 81, 405, 225, 30, 1;
(End)
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016
MATHEMATICA
Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
PROG
(PARI) for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 02 2002
STATUS
approved