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

1,2

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.

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

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

Cf. A075503, A075505.

Columns 2-7 are A076008-A076013.

Sequence in context: A318935 A347490 A038291 * A373628 A138342 A101678

Adjacent sequences: A075501 A075502 A075503 * A075505 A075506 A075507

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 02 2002

Status

approved