|
Search: id:A075503
|
|
|
| A075503 |
|
Stirling2 triangle with scaled diagonals (powers of 8). |
|
+0
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; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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(a(n,m)x^m,m=1..n), n>=1, have e.g.f. J(x; z)= exp((exp(8*z)-1)*x/8)-1.
Row sums give A075507(n),n>=1. The columns (without leading zeros) give A001018 (powers of 8), A060195, A076003-A076007 for m=1..7.
|
|
FORMULA
|
a(n, m)=(8^(n-m))S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=sum((A075513(m, p)*((p+1)*8)^(n-m))/(m-1)!, p=0..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(1-8k*x, k=1..m), 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).
|
|
CROSSREFS
|
Cf. A075502, A075504.
Sequence in context: A089276 A051932 A038279 this_sequence A051379 A143499 A114152
Adjacent sequences: A075500 A075501 A075502 this_sequence A075504 A075505 A075506
|
|
KEYWORD
|
nonn,easy,tabl
|
|
AUTHOR
|
Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
|
|
|
Search completed in 0.002 seconds
|
