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A051142
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Generalized Stirling number triangle of first kind.
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12
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1, -4, 1, 32, -12, 1, -384, 176, -24, 1, 6144, -3200, 560, -40, 1, -122880, 70144, -14400, 1360, -60, 1, 2949120, -1806336, 415744, -47040, 2800, -84, 1, -82575360, 53526528, -13447168, 1732864, -125440, 5152, -112, 1, 2642411520, -1795424256, 483835904
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OFFSET
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1,2
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COMMENTS
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a(n,m) = R_n^m(a=0, b=4) in the notation of the given 1961 and 1962 references.
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 - 4*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 diagonal d >= 0 (main diagonal d = 0) scaled with 4^d.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+4) (A047053) giving unsigned values and adding 1, 0, 0, 0, ... as column 0. For the definition of the Bell transform, see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
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LINKS
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FORMULA
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a(n, m) = a(n-1, m-1) - 4*(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 + 4*x)/4)^m/m!.
a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
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EXAMPLE
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Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-4, 1;
32, -12, 1;
-384, 176, -24, 1;
6144, -3200, 560, -40, 1,
-122880, 70144, -14400, 1360, -60, 1;
...
3rd row o.g.f.: E(3,x) = 32*x - 12*x^2 + x^3.
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MATHEMATICA
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Table[StirlingS1[n, m] 4^(n - m), {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
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PROG
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(Sage) # uses[bell_transform from A264428]
# Unsigned values and an additional first column (1, 0, 0, 0, ...).
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
mfact = [multifact_4_4(k) for k in (0..n)]
return bell_transform(n, mfact)
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CROSSREFS
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First (m=1) column sequence is: A047053(n-1).
Row sums (signed triangle): A008545(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A007696(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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