A051150 - OEIS

A051150

Generalized Stirling number triangle of first kind.

%I #19 Jun 09 2020 06:55:51

%S 1,-5,1,50,-15,1,-750,275,-30,1,15000,-6250,875,-50,1,-375000,171250,

%T -28125,2125,-75,1,11250000,-5512500,1015000,-91875,4375,-105,1,

%U -393750000,204187500,-41037500,4230625

%N Generalized Stirling number triangle of first kind.

%C a(n,m) = R_n^m(a=0, b=5) 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 - 5*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 5^d.

%H Wolfdieter Lang, <a href="/A051150/a051150.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) - 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 E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.

%F a(n, m) = S1(n, m)*5^(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 -5, 1;

%e 50, -15, 1;

%e -750, 275, -30, 1;

%e 15000, -6250, 875, -50, 1;

%e -375000, 171250, -28125, 2125, -75, 1;

%e ...

%e 3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.

%Y First (m=1) column sequence: A052562(n-1).

%Y Row sums (signed triangle): A008546(n-1)*(-1)^(n-1).

%Y Row sums (unsigned triangle): A008548(n).

%Y Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4).

%K sign,easy,tabl

%O 1,2

%A _Wolfdieter Lang_