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A051186 Generalized Stirling number triangle of first kind. 6
1, -7, 1, 98, -21, 1, -2058, 539, -42, 1, 57624, -17150, 1715, -70, 1, -2016840, 657874, -77175, 4165, -105, 1, 84707280, -29647548, 3899224, -252105, 8575, -147, 1, -4150656720, 1537437132, -220709524, 16252369 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n,m)= R_n^m(a=0,b=7) 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-7*j,j=0..n-1), n >= 1, E(0,x) := 1, are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

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..32.

W. Lang, First ten rows.

FORMULA

a(n, m) = a(n-1, m-1) - 7*(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+7*x))/7)^m)/m!.

a(n,m)=((7^(n-m))*S1(n,m) with the (signed) Stirling1 triangle S1(n,m)=A008275(n,m).

EXAMPLE

{1}; {-7,1}; {98,-21,1}; {-2058,539,-42,1}; ... E(3,x) = 98*x-21*x^2+x^3.

CROSSREFS

First (m=1) column sequence is: A051188(n-1). Row sums (signed triangle): A049209(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A045754(n). The b=1..6 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151.

Sequence in context: A027517 A092082 A013559 * A012034 A138324 A052122

Adjacent sequences:  A051183 A051184 A051185 * A051187 A051188 A051189

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified January 22 16:47 EST 2019. Contains 319364 sequences. (Running on oeis4.)