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(10*z) - 1)*x/10) - 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
FORMULA
a(n, m) = (10^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*10)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 10m*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} (1-10k*x), m >= 1.
E.g.f. for m-th column: (((exp(10x)-1)/10)^m)/m!, m >= 1.
EXAMPLE
[1]; [10,1]; [100,30,1]; ...; p(3,x) = x(100 + 30*x + x^2).
From Andrew Howroyd, Mar 25 2017: (Start)
Triangle starts
* 1
* 10 1
* 100 30 1
* 1000 700 60 1
* 10000 15000 2500 100 1
* 100000 310000 90000 6500 150 1
* 1000000 6300000 3010000 350000 14000 210 1
* 10000000 127000000 96600000 17010000 1050000 26600 280 1
(End)
MATHEMATICA
Flatten[Table[10^(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(10^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 02 2002
STATUS
approved