A075500 Stirling2 triangle with scaled diagonals (powers of 5).

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

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(5*z) - 1)*x/5) - 1.

Examples

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)
		

Links

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

Crossrefs

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Mathematica

Flatten[Table[5^(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[5^#&, 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(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (5^(n-m)) * stirling2(n, m).

a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.

a(n, m) = 5m*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}(1-5k*x), m >= 1.

E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.