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A075503 Stirling2 triangle with scaled diagonals (powers of 8). 9
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 (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(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: A089276 A051932 A038279 * A260040 A051379 A143499

Adjacent sequences:  A075500 A075501 A075502 * A075504 A075505 A075506

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 02 2002

STATUS

approved

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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)