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A051231
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Generalized Stirling number triangle of the first kind.
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2
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1, -9, 1, 162, -27, 1, -4374, 891, -54, 1, 157464, -36450, 2835, -90, 1, -7085880, 1797714, -164025, 6885, -135, 1, 382637520, -104162436, 10655064, -535815, 14175, -189, 1, -24106163760, 6944870988, -775431468, 44411409, -1428840, 26082, -252, 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=9) 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 - 9*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962). Special cases were tabulated in this and other related papers.
Special cases of these numbers are related to numbers introduced by Nörlund (1924).
These numbers are defined via the g.f. 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, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let 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) which satisfy Product_{r=0}^{n-1} (x - r) = Sum_{m=0..n} S1(n,m)*x^m with S1(n,n) = 1 for n >= 0, S1(n,0) = 0 for n >= 1, and S1(n, m) = 0 for m > n. (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
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=9) 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) - 9*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, 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 + 9*x)/9)^m/m!.
T(n,m) = 9^(n-m)*Stirling1(n,m) = 9^(n-m)*A048994(n,m) = 9^(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/9)*log(1 + 9*x)) - 1 = (1 + 9*x)^(y/9) - 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;
-9, 1;
162, -27, 1;
-4374, 891, -54, 1;
157464, -36450, 2835, -90, 1;
-7085880, 1797714, -164025, 6885, -135, 1;
...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x-9*j) = 162*x - 27*x^2 + x^3. [Edited by Petros Hadjicostas, Jun 06 2020]
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CROSSREFS
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First (m=1) column sequence is A051232(n-1).
Row sums (signed triangle): A049211(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045756(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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