OFFSET
1,2
COMMENTS
This is a lower triangular infinite matrix of the Jabotinsky type. See the 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(6*z) - 1)*x/6) - 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
a(n, m) = (6^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 6m*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}(1-6k*x), m >= 1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.
EXAMPLE
[1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).
From Andrew Howroyd, Mar 25 2017: (Start)
Triangle starts
* 1
* 6 1
* 36 18 1
* 216 252 36 1
* 1296 3240 900 60 1
* 7776 40176 19440 2340 90 1
* 46656 489888 390096 75600 5040 126 1
* 279936 5925312 7511616 2204496 226800 9576 168 1
(End)
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 6^n, 9); # Peter Luschny, Jan 28 2016
MATHEMATICA
Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[6^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
PROG
(PARI) for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 02 2002
STATUS
approved