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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 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-6*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

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). A008275 (Stirling1 triangle) for b=1, A039683 for b=2, b=3: A051141, b=4: A051142, b=5: A051150.

This is the signed Stirling1 triangle A008275 with diagonal d>=0 (main diagonal d=0) scaled with 6^d.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..35.

W. Lang, First 10 rows.

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, a(1, 1)=1. E.g.f. for m-th column of 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

{1}; {-6,1}; {72,-18,1}; {-1296,396,-36,1}; ...

E(3,x) = 72*x-18*x^2+x^3.

CROSSREFS

Sequence in context: A266364 A009384 A280520 * A009330 A300512 A318107

Adjacent sequences:  A051148 A051149 A051150 * A051152 A051153 A051154

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)