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A075505 Stirling2 triangle with scaled diagonals (powers of 10). 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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

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Last modified September 25 00:02 EDT 2020. Contains 337333 sequences. (Running on oeis4.)