A075501 Stirling2 triangle with scaled diagonals (powers of 6).

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, 1679616, 71383680

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(6*z) - 1)*x/6) - 1.

Examples

			[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)
		

Links

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

Crossrefs

Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.

Cf. A075500, A075502.

Programs

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 *)

PARI

for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

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.