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A051150 Generalized Stirling number triangle of first kind.

%I A051150 #19 Jun 09 2020 06:55:51
%S A051150 1,-5,1,50,-15,1,-750,275,-30,1,15000,-6250,875,-50,1,-375000,171250,
%T A051150 -28125,2125,-75,1,11250000,-5512500,1015000,-91875,4375,-105,1,
%U A051150 -393750000,204187500,-41037500,4230625
%N A051150 Generalized Stirling number triangle of first kind.
%C A051150 a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.
%C A051150 a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 5*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C A051150 This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.
%H A051150 Wolfdieter Lang, <a href="/A051150/a051150.txt">First 10 rows</a>.
%H A051150 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. [The numbers R_n^m(a,b) are first introduced.]
%H A051150 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. [Special cases of the numbers R_n^m(a,b) are tabulated.]
%F A051150 a(n, m) = a(n-1, m-1) - 5*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
%F A051150 E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.
%F A051150 a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
%e A051150 Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
%e A051150         1;
%e A051150        -5,      1;
%e A051150        50,    -15,      1;
%e A051150      -750,    275,    -30,   1;
%e A051150     15000,  -6250,    875, -50,    1;
%e A051150   -375000, 171250, -28125, 2125, -75, 1;
%e A051150   ...
%e A051150 3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.
%Y A051150 First (m=1) column sequence: A052562(n-1).
%Y A051150 Row sums (signed triangle): A008546(n-1)*(-1)^(n-1).
%Y A051150 Row sums (unsigned triangle): A008548(n).
%Y A051150 Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4).
%K A051150 sign,easy,tabl
%O A051150 1,2
%A A051150 _Wolfdieter Lang_