A051151 Generalized Stirling number triangle of first kind.

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

Offset: 1

Author

Wolfdieter Lang

Keywords

• sign
• easy
• tabl

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.

Examples

			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.
		

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 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).

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).

Extensions

Various sections edited by Petros Hadjicostas, Jun 08 2020