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A075501 Stirling2 triangle with scaled diagonals (powers of 6). 9
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 (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(6*z) - 1)*x/6) - 1.

LINKS

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

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.

EXAMPLE

[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)

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-Fran├žois Alcover, Jun 23 2018, after Peter Luschny *)

PROG

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

CROSSREFS

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

Cf. A075500, A075502.

Sequence in context: A051930 A147320 A038255 * A089504 A145927 A113365

Adjacent sequences:  A075498 A075499 A075500 * A075502 A075503 A075504

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 02 2002

STATUS

approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)