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 A051151 Generalized Stirling number triangle of first kind. 7
 1, -6, 1, 72, -18, 1, -1296, 396, -36, 1, 31104, -10800, 1260, -60, 1, -933120, 355104, -48600, 3060, -90, 1, 33592320, -13716864, 2104704, -158760, 6300, -126, 1, -1410877440, 609700608, -102114432, 8772624, -423360, 11592, -168 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references. 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). This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d. LINKS Wolfdieter Lang, First 10 rows. D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are first introduced.] D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, 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.] FORMULA 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. E.g.f. for the m-th column of the signed triangle: (log(1 + 6*x)/6)^m)/m!. a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle). EXAMPLE Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:         1;        -6,      1;        72,    -18,      1;     -1296,    396,    -36,    1;     31104, -10800,   1260,  -60,   1;   -933120, 355104, -48600, 3060, -90, 1;    ... 3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3. CROSSREFS First (m=1) column sequence is: A047058(n-1). Row sums (signed triangle): A008543(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A008542(n). Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4), A051150 (b=5). Sequence in context: A266364 A009384 A280520 * A009330 A300512 A318107 Adjacent sequences:  A051148 A051149 A051150 * A051152 A051153 A051154 KEYWORD sign,easy,tabl AUTHOR EXTENSIONS Various sections edited by Petros Hadjicostas, Jun 08 2020 STATUS approved

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Last modified September 24 21:35 EDT 2020. Contains 337322 sequences. (Running on oeis4.)