A051150 Generalized Stirling number triangle of first kind.

1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625

Offset: 1

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

Comments

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

Examples

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -5,      1;
       50,    -15,      1;
     -750,    275,    -30,   1;
    15000,  -6250,    875, -50,    1;
  -375000, 171250, -28125, 2125, -75, 1;
  ...
3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.
		

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.]

Crossrefs

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

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

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

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

Formula

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.

E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.

a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).