A075910 -id:A075910 - cp's OEIS frontend

A075499 Stirling2 triangle with scaled diagonals (powers of 4).

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

Offset: 1

Wolfdieter Lang, Oct 02 2002

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

			[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)

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

Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.

Cf. A075498, A075500.

Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from 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

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.

A075909 Sixth column of triangle A075499.

Original entry on oeis.org

1, 84, 4256, 169344, 5843712, 183794688, 5421678592, 152720375808, 4157366140928, 110282217357312, 2867778350481408, 73424436820180992, 1857023919127527424, 46511918954689069056, 1155904251854380335104

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..5} A075513(6,m)*exp(4*(m+1)*x)/5!.

Cf. A075908, A075910.

a(n) = A075499(n+6, 6) = (4^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*4)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 4*k*x).
E.g.f.: (d^6/dx^6)(((exp(4*x)-1)/4)^6)/6! = (-exp(4*x) + 160*exp(8*x) - 2430*exp(12*x) + 10240*exp(16*x) - 15625*exp(20*x) + 7776*exp(24*x))/5!.

cp's OEIS Frontend

A075499 Stirling2 triangle with scaled diagonals (powers of 4).

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