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%I #13 Jun 22 2018 09:22:37
%S 1,-10,1,200,-30,1,-6000,1100,-60,1,240000,-50000,3500,-100,1,
%T -12000000,2740000,-225000,8500,-150,1,720000000,-176400000,16240000,
%U -735000,17500,-210,1,-50400000000,13068000000,-1313200000,67690000,-1960000,32200,-280,1,4032000000000,-1095840000000
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= R_n^m(a=0,b=10) in the notation of the given reference.
%C 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-10*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C Also the Bell transform of the sequence (-1)^n*A051262(n) without column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%D 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.
%F a(n, m) = a(n-1, m-1) - 10*(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: (((log(1+10*x))/10)^m)/m!.
%e {1}; {-10,1}; {200,-30,1}; {-6000,1100,-60,1}; ... E(3,x) = 200*x-30*x^2+x^3.
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> (-1)^n*n!*10^n, 9); # _Peter Luschny_, Jan 28 2016
%t rows = 9;
%t t = Table[(-1)^n*n!*10^n, {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%Y First (m=1) (unsigned) column sequence is: A051262(n-1). Row sums (signed triangle): A049212(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045757(n). b=8: A051187, b=9: A051231.
%K sign,easy,tabl
%O 1,2
%A _Wolfdieter Lang_
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