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A075504
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Stirling2 triangle with scaled diagonals (powers of 9).
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9
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1, 9, 1, 81, 27, 1, 729, 567, 54, 1, 6561, 10935, 2025, 90, 1, 59049, 203391, 65610, 5265, 135, 1, 531441, 3720087, 1974861, 255150, 11340, 189, 1, 4782969, 67493007, 57041334, 11160261, 765450
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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(a(n,m)x^m,m=1..n), n>=1, have e.g.f. J(x; z)= exp((exp(9*z)-1)*x/9)-1.
Row sums give A075508(n),n>=1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.
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FORMULA
| a(n, m)=(9^(n-m))S2(n, m) with S2(n, m) := A008277(n, m) (Stirling2).
a(n, m)=sum((A075513(m, p)*((p+1)*9)^(n-m))/(m-1)!, p=0..m-1) for n>=m>=1 else 0.
a(n, m)=9m*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(1-9k*x, k=1..m), m>=1.
E.g.f. for m-th column: (((exp(9x)-1)/9)^m)/m!, m>=1.
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EXAMPLE
| [1]; [9,1]; [81,27,1]; ...; p(3,x)=x(81+27*x+x^2).
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CROSSREFS
| Cf. A075503, A075505.
Sequence in context: A107892 A050303 A038291 * A138342 A101678 A051380
Adjacent sequences: A075501 A075502 A075503 * A075505 A075506 A075507
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
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