A075503 Stirling2 triangle with scaled diagonals (powers of 8).

1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 1

Offset: 1

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• tabl

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(8*z) - 1)*x/8) - 1.

Examples

			[1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       8        1
*      64       24        1
*     512      448       48       1
*    4096     7680     1600      80      1
*   32768   126976    46080    4160    120     1
*  262144  2064384  1232896  179200   8960   168   1
* 2097152 33292288 31653888 6967296 537600 17024 224 1
(End)
		

Links

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

Crossrefs

Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.

Cf. A075502, A075504.

Programs

Mathematica

Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

PARI

for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (8^(n-m)) * stirling2(n, m).

a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.

a(n, m) = 8m*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}(1-8k*x), m >= 1.

E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.