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A051151 Generalized Stirling number triangle of first kind.

%I A051151 #27 Jan 04 2025 16:34:58
%S A051151 1,-6,1,72,-18,1,-1296,396,-36,1,31104,-10800,1260,-60,1,-933120,
%T A051151 355104,-48600,3060,-90,1,33592320,-13716864,2104704,-158760,6300,
%U A051151 -126,1,-1410877440,609700608,-102114432,8772624,-423360,11592,-168
%N A051151 Generalized Stirling number triangle of first kind.
%C A051151 a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.
%C A051151 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 A051151 This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.
%H A051151 Wolfdieter Lang, <a href="/A051151/a051151.txt">First 10 rows</a>.
%H A051151 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 A051151 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 A051151 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 A051151 E.g.f. for the m-th column of the signed triangle: ((log(1 + 6*x)/6)^m)/m!.
%F A051151 a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
%e A051151 Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
%e A051151         1;
%e A051151        -6,      1;
%e A051151        72,    -18,      1;
%e A051151     -1296,    396,    -36,    1;
%e A051151     31104, -10800,   1260,  -60,   1;
%e A051151   -933120, 355104, -48600, 3060, -90, 1;
%e A051151    ...
%e A051151 3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3.
%Y A051151 First (m=1) column sequence is: A047058(n-1).
%Y A051151 Row sums (signed triangle): A008543(n-1)*(-1)^(n-1).
%Y A051151 Row sums (unsigned triangle): A008542(n).
%Y A051151 Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5).
%K A051151 sign,easy,tabl
%O A051151 1,2
%A A051151 _Wolfdieter Lang_
%E A051151 Various sections edited by _Petros Hadjicostas_, Jun 08 2020