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A051187
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Generalized Stirling number triangle of the first kind.
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5
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1, -8, 1, 128, -24, 1, -3072, 704, -48, 1, 98304, -25600, 2240, -80, 1, -3932160, 1122304, -115200, 5440, -120, 1, 188743680, -57802752, 6651904, -376320, 11200, -168, 1, -10569646080, 3425697792, -430309376, 27725824, -1003520, 20608, -224, 1
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OFFSET
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1,2
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COMMENTS
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T(n,m)= R_n^m(a=0, b=8) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 8*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962). Such numbers are related to the work of Nörlund (1924).
They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m).
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=8) but with no zero row or column. (End)
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LINKS
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D. S. Mitrinovic and R. S. Mitrinovic, Sur les polynômes de Stirling, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
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FORMULA
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T(n, m) = T(n-1, m-1) - 8*(n-1)*T(n-1, m) for n >= m >= 1; T(n, m) := 0 for n < m; T(n, 0) := 0 for n >= 1; T(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 8*x)/8)^m/m!.
T(n,m) = 8^(n-m)*Stirling1(n,m) = 8^(n-m)*A048994(n,m) = 8^(n-m)*A008275(n,m) for n >= m >= 1.
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/8)*log(1 + 8*x)) - 1 = (1 + 8*x)^(y/8) - 1. (End)
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EXAMPLE
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Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-8, 1;
128, -24, 1;
-3072, 704, -48, 1;
98304, -25600, 2240, -80, 1;
-3932160, 1122304, -115200, 5440, -120, 1;
188743680, -57802752, 6651904, -376320, 11200, -168, 1;
...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 8*j) = 128*x - 24*x^2 + x^3.
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CROSSREFS
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First (m=1) column sequence is: A051189(n-1).
Row sums (signed triangle): A049210(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045755(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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