A075505 Stirling2 triangle with scaled diagonals (powers of 10).

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

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(10*z) - 1)*x/10) - 1.

Examples

			[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)
		

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.

Crossrefs

Row sums are A075509.

Cf. A075504.

Programs

Mathematica

Flatten[Table[10^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

PARI

for(n=1, 11, for(m=1, n, print1(10^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

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.