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

1,2

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

Links

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

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.

Example

[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)

Mathematica

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

Prog

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

Crossrefs

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

Cf. A075502, A075504.

Sequence in context: A051932 A347489 A038279 * A260040 A051379 A143499

Adjacent sequences: A075500 A075501 A075502 * A075504 A075505 A075506

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 02 2002

Status

approved