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A051150
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Generalized Stirling number triangle of first kind.
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8
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1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n,m)= R_n^m(a=0,b=5) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-5*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
First (m=1) column sequence is: A052562(n-1). Row sums (signed triangle): A008546(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A008548(n). A008275 (Stirling1 triangle) for b=1, A039683 for b=2, b=3: A051141, b=4: A051142.
This is the signed Stirling1 triangle A008275 with diagonal d>=0 (main diagonal d=0) scaled with 5^d.
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REFERENCES
| Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
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LINKS
| W. Lang, First 10 rows.
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FORMULA
| a(n, m) = a(n-1, m-1) - 5*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+5*x))/5)^m)/m!.
a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
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EXAMPLE
| {1}; {-5,1}; {50,-15,1}; {-750,275,-30,1}; ...
E(3,x) = 50*x-15*x^2+x^3.
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CROSSREFS
| Sequence in context: A134273 A048897 A049029 * A144341 A144342 A144268
Adjacent sequences: A051147 A051148 A051149 * A051151 A051152 A051153
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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