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%I #23 Jun 08 2020 17:03:40
%S 1,-6,1,72,-18,1,-1296,396,-36,1,31104,-10800,1260,-60,1,-933120,
%T 355104,-48600,3060,-90,1,33592320,-13716864,2104704,-158760,6300,
%U -126,1,-1410877440,609700608,-102114432,8772624,-423360,11592,-168
%N Generalized Stirling number triangle of first kind.
%C a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.
%C 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-6*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%C This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.
%H Wolfdieter Lang, <a href="/A051151/a051151.txt">First 10 rows</a>.
%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. [The numbers R_n^m(a,b) are first introduced.]
%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. [Special cases of the numbers R_n^m(a,b) are tabulated.]
%F a(n, m) = a(n-1, m-1) - 6*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
%F E.g.f. for the m-th column of the signed triangle: (log(1 + 6*x)/6)^m)/m!.
%F a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
%e Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
%e 1;
%e -6, 1;
%e 72, -18, 1;
%e -1296, 396, -36, 1;
%e 31104, -10800, 1260, -60, 1;
%e -933120, 355104, -48600, 3060, -90, 1;
%e ...
%e 3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3.
%Y First (m=1) column sequence is: A047058(n-1).
%Y Row sums (signed triangle): A008543(n-1)*(-1)^(n-1).
%Y Row sums (unsigned triangle): A008542(n).
%Y Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5).
%K sign,easy,tabl
%O 1,2
%A _Wolfdieter Lang_
%E Various sections edited by _Petros Hadjicostas_, Jun 08 2020
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