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A051141
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Generalized Stirling number triangle of first kind.
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11
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1, -3, 1, 18, -9, 1, -162, 99, -18, 1, 1944, -1350, 315, -30, 1, -29160, 22194, -6075, 765, -45, 1, 524880, -428652, 131544, -19845, 1575, -63, 1, -11022480, 9526572, -3191076, 548289, -52920, 2898, -84, 1, 264539520, -239660208
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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]
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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.
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LINKS
| W. Lang, First 10 rows.
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FORMULA
| 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).
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EXAMPLE
| {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)
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MATHEMATICA
| a[n_, m_] /; n >= m >= 1 := a[n, m] = a[n-1, m-1] - 3(n-1)*a[n-1, m]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]] (* From Jean-François Alcover, Jun 1 2011, after formula *)
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CROSSREFS
| 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 * A068141 A051238 A120984
Adjacent sequences: A051138 A051139 A051140 * A051142 A051143 A051144
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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