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A051142
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Generalized Stirling number triangle of first kind.
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11
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1, -4, 1, 32, -12, 1, -384, 176, -24, 1, 6144, -3200, 560, -40, 1, -122880, 70144, -14400, 1360, -60, 1, 2949120, -1806336, 415744, -47040, 2800, -84, 1, -82575360, 53526528, -13447168, 1732864, -125440, 5152, -112, 1, 2642411520, -1795424256, 483835904
(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=4) 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-4*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle with diagonal d>=0 (main diagonal d=0) scaled with 4^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) - 4*(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+4*x))/4)^m)/m!.
a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
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EXAMPLE
| {1}; {-4,1}; {32,-12,1}; {-384,176,-24,1}; ...
E(3,x) = 32*x-12*x^2+x^3.
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CROSSREFS
| First (m=1) column sequence is: A047053(n-1). Row sums (signed triangle): A008545(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A007696(n). Cf. A008275 (Stirling1 triangle), for b=1, A039683 for b=2. Cf. A051141.
Sequence in context: A190647 A123126 A174501 * A075804 A059844 A144284
Adjacent sequences: A051139 A051140 A051141 * A051143 A051144 A051145
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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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