A075504 Stirling2 triangle with scaled diagonals (powers of 9).

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

Offset: 1

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• tabl

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.

Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

Examples

			[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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Cf. A075503, A075505.

Columns 2-7 are A076008-A076013.

Programs

Mathematica

Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

PARI

for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

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.