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(9*z) - 1)*x/9) - 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
FORMULA
a(n, m) = (9^(n-m)) * stirling2(n, m).
a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 9m*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}(1-9k*x), m >= 1.
E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.
EXAMPLE
[1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
From Andrew Howroyd, Mar 25 2017: (Start)
Triangle starts
* 1
* 9 1
* 81 27 1
* 729 567 54 1
* 6561 10935 2025 90 1
* 59049 203391 65610 5265 135 1
* 531441 3720087 1974861 255150 11340 189 1
* 4782969 67493007 57041334 11160261 765450 21546 252 1
(End)
MATHEMATICA
Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
PROG
(PARI) for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 02 2002
STATUS
approved