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

1,2

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

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.

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.

Example

[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)

Mathematica

Flatten[Table[10^(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(10^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017

Crossrefs

Row sums are A075509.

Cf. A075504.

Sequence in context: A165293 A038303 A178870 * A130310 A333685 A288050

Adjacent sequences: A075502 A075503 A075504 * A075506 A075507 A075508

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 02 2002

Status

approved