|
| |
|
|
A075499
|
|
Stirling2 triangle with scaled diagonals (powers of 4).
|
|
14
|
|
|
|
1, 4, 1, 16, 12, 1, 64, 112, 24, 1, 256, 960, 400, 40, 1, 1024, 7936, 5760, 1040, 60, 1, 4096, 64512, 77056, 22400, 2240, 84, 1, 16384, 520192, 989184, 435456, 67200, 4256, 112, 1, 65536, 4177920, 12390400, 7956480, 1779456, 169344, 7392, 144, 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(4*z) - 1)*x/4) - 1
Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
|
|
|
LINKS
|
Andrew Howroyd, Table of n, a(n) for n = 1..1275
|
|
|
FORMULA
|
a(n, m) = (4^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 4m*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}(1-4k*x), m >= 1.
E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1.
|
|
|
EXAMPLE
|
[1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2).
From Andrew Howroyd, Mar 25 2017: (Start)
Triangle starts
* 1
* 4 1
* 16 12 1
* 64 112 24 1
* 256 960 400 40 1
* 1024 7936 5760 1040 60 1
* 4096 64512 77056 22400 2240 84 1
* 16384 520192 989184 435456 67200 4256 112 1
(End)
|
|
|
MATHEMATICA
|
Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
|
|
|
PROG
|
(Sage)
# The function inverse_bell_transform is defined in A265605.
# Adds a column 1, 0, 0, ... at the left side of the triangle.
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
inverse_bell_matrix(multifact_4_4, 9) # Peter Luschny, Dec 31 2015
(PARI)
for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017
|
|
|
CROSSREFS
|
Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.
Cf. A075498, A075500.
Sequence in context: A271262 A292922 A117438 * A099394 A269698 A059991
Adjacent sequences: A075496 A075497 A075498 * A075500 A075501 A075502
|
|
|
KEYWORD
|
nonn,easy,tabl
|
|
|
AUTHOR
|
Wolfdieter Lang, Oct 02 2002
|
|
|
STATUS
|
approved
|
| |
|
|
