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%I #65 Jun 10 2020 09:23:17
%S 1,-9,1,162,-27,1,-4374,891,-54,1,157464,-36450,2835,-90,1,-7085880,
%T 1797714,-164025,6885,-135,1,382637520,-104162436,10655064,-535815,
%U 14175,-189,1,-24106163760,6944870988,-775431468,44411409,-1428840,26082,-252,1
%N Generalized Stirling number triangle of the first kind.
%C T(n,m) = R_n^m(a=0, b=9) in the notation of the given 1962 reference.
%C 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).
%C From _Petros Hadjicostas_, Jun 06 2020: (Start)
%C 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.
%C Special cases of these numbers are related to numbers introduced by Nörlund (1924).
%C 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.)
%C 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.)
%C 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.
%C For the current array, T(n,m) = R_n^m(a=0, b=9) but with no zero row or column. (End)
%H D. S. Mitrinovic, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k750m/f1360.image.r=Mitrinovic">Sur une relation de récurrence relative aux nombres de Bernoulli</a>, Comptes rendus de l'Académie des sciences de Paris, t. 250 (1960), 4266-4267.
%H D. S. Mitrinovic, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k762d/f996.image.r=1961%20mitrinovic">Sur une classe de nombres reliés aux nombres de Stirling</a>, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://gdz.sub.uni-goettingen.de/id/PPN311570321_0010?tify={%22pages%22:[45],%22view%22:%22info%22}">Sur les polynômes de Stirling</a>, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667599">Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieux</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667490">Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667130">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43669525">Tableaux d'une classe de nombres reliés aux nombres de Stirling. V</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 132/142 (1965), 1-22.
%H Niels Nörlund, <a href="https://eudml.org/doc/204170">Vorlesungen über Differenzenrechnung</a>, Springer, Berlin, 1924.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragoslav_Mitrinovi%C4%87">Dragoslav Mitrinovic</a>.
%F 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.
%F E.g.f. for the m-th column of the signed triangle: (log(1 + 9*x)/9)^m/m!.
%F From _Petros Hadjicostas_, Jun 07 2020: (Start)
%F 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.
%F 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)
%e Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
%e 1;
%e -9, 1;
%e 162, -27, 1;
%e -4374, 891, -54, 1;
%e 157464, -36450, 2835, -90, 1;
%e -7085880, 1797714, -164025, 6885, -135, 1;
%e ...
%e 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]
%Y First (m=1) column sequence is A051232(n-1).
%Y Row sums (signed triangle): A049211(n-1)*(-1)^(n-1).
%Y Row sums (unsigned triangle): A045756(n).
%Y Cf. A008275 (b=1 triangle), A048994 (b=1 triangle), A051187 (b=8 triangle).
%K sign,easy,tabl
%O 1,2
%A _Wolfdieter Lang_
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