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A119724
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Generalized Pascal's triangle made using Mod[(Prime[n] - 1)/2, 4] == 2 primorial-like Stirling polynomials.
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1
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1, 1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -5, 10, -10, 5, -1, 1, -10, 35, -60, 55, -26, 5, 1, -15, 85, -235, 355, -301, 135, -25, 1, -20, 160, -660, 1530, -2076, 1640, -700, 125, 1, -25, 260, -1460, 4830, -9726, 12020, -8900, 3625, -625, 1, -30, 385, -2760, 12130, -33876, 60650, -69000, 48125, -18750
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OFFSET
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0,5
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COMMENTS
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Apparently the sequence is based on the list of primes p=5, 13, 29, 37, 53, 61,... for which (p-1)/2 == 2 (mod 4), derived from A005097. The coefficients of the polynomial of degree n are listed in row n, where the polynomial is a product of the form prod_i (1-p_i*x), and p_i is the largest prime of that modular subset which is less than i. - R. J. Mathar, May 15 2013
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LINKS
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FORMULA
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a(n) = Flatten[Join[{{1}}, Table[Reverse[CoefficientList[Product[x - p1[n], {n, 0, m}], x]], {m, 0, 10}]]]
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EXAMPLE
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1; # 1
1, -1; # 1-x
1, -2, 1; # (1-x)^2
1, -3, 3, -1; # (1-x)^3
1, -4, 6, -4, 1; # (1-x)^4
1, -5, 10, -10, 5, -1; # (1-x)^5
1, -10, 35, -60, 55, -26, 5; # (1-x)^5*(1-5x)
1, -15, 85, -235, 355, -301, 135, -25; # (1-x)^5*(1-5x)^2
1, -20, 160, -660, 1530, -2076, 1640, -700, 125; # (1-x)^5*(1-5x)^3
1, -25, 260, -1460, 4830, -9726, 12020, -8900, 3625, -625; # (1-x)^5*(1-5x)^4
1, -30, 385, -2760, 12130, -33876, 60650, -69000, 48125, -18750,.. # (1-x)^5*(1-5x)^5
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MATHEMATICA
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a = Join[{{1}}, Table[Reverse[ CoefficientList[Product[x - p1[n], {n, 0, m}], x]], {m, 0, 10}]] aout = Flatten[a]
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CROSSREFS
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KEYWORD
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sign,uned,tabf,obsc
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AUTHOR
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EXTENSIONS
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Should be edited in the same way that I edited A118686. Unfortunately p1 has not been defined, but must be related to "Mod[(Prime[n] - 1)/2, 4] == 2". Compare the definition of p[n] in A118686. - N. J. A. Sloane, Oct 08 2006
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STATUS
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approved
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