1,2

a(n,m)= R_n^m(a=0,b=3) in the notation of the given reference.

a(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n) = product(x-3*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

This is the signed Stirling1 triangle with diagonals d>=0 (main diagonal d=0) scaled with 3^d.

Exponential Riordan array [1/(1+3x),ln(1+3x)/3]. The unsigned triangle is [1/(1-3x),ln(1/(1-3x)^(1/3))]. [From Paul Barry (pbarry(AT)wit.ie), Apr 29 2009]

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

W. Lang, First 10 rows.

a(n, m) = a(n-1, m-1) - 3*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+3*x))/3)^m)/m!.

a(n, m) = S1(n, m)*3^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

{1}; {-3,1}; {18,-9,1}; {-162,99,-18,1}; ...

E(3,x) = 18*x-9*x^2+x^3.

Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 29 2009: (Start)

The unsigned array [1/(1-3x),ln(1/(1-3x)^(1/3))] has production matrix

3, 1,

9, 6, 1,

27, 27, 9, 1,

81, 108, 54, 12, 1,

243, 405, 270, 90, 15, 1,

729, 1458, 1215, 540, 135, 18, 1

which is A007318^{3} beheaded. (End)

First (m=1) column sequence is: A032031(n-1). Row sums (signed triangle): A008544(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A007559(n). Cf. A008275 (Stirling1 triangle), for b=1, A039683 for b=2. Cf. A051142.

Sequence in context: A143849 A105626 A071210 this_sequence A068141 A051238 A120984

Adjacent sequences: A051138 A051139 A051140 this_sequence A051142 A051143 A051144

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)