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A051141
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Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
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14
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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
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OFFSET
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1,2
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COMMENTS
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Previous name was: Generalized Stirling number triangle of first kind.
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_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 3*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 with diagonals d>=0 (main diagonal d=0) scaled with 3^d.
Exponential Riordan array [1/(1 + 3*x), log(1 + 3*x)/3]. The unsigned triangle is [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))]. - Paul Barry, Apr 29 2009
Also the Bell transform of the triple factorial numbers A032031 which adds a first column (1, 0, 0 ...) on the left side of the triangle and computes the unsigned values. For the definition of the Bell transform, see A264428. See A004747 for the triple factorial numbers A008544 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015
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LINKS
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FORMULA
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a(n, m) = a(n-1, m-1) - 3*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) = 0 for 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 + 3*x)/3)^m/m!.
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EXAMPLE
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Triangle starts:
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;
---
Row polynomial E(3,x) = 18*x-9*x^2+x^3.
The unsigned array [1/(1 - 3*x), log(1/(1 - 3*x)^(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 (by viewing A007318 as a lower triangular matrix). See the comment above. (End)
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MATHEMATICA
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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]] (* Jean-François Alcover, Jun 01 2011, after formula *)
Table[StirlingS1[n, m]*3^(n - m), {n, 1, 10}, {m, 1, n}]//Flatten (* G. C. Greubel, Oct 24 2017 *)
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PROG
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(Sage) # uses[bell_transform from A264428]
triplefactorial = lambda n: 3^n*factorial(n)
trifact = [triplefactorial(k) for k in (0..n)]
return bell_transform(n, trifact)
(PARI) for(n=1, 10, for(m=1, n, print1(stirling(n, m, 1)*3^(n-m), ", "))) \\ G. C. Greubel, Oct 24 2017
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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EXTENSIONS
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Name clarified using a formula of the author by Peter Luschny, Dec 23 2015
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STATUS
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approved
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