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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

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Last modified March 18 13:47 EDT 2019. Contains 321289 sequences. (Running on oeis4.)