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A327855
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Irregular triangle read by rows: T(n,k) = [x^k] (Sum_{i=0..prime(n)-1} (1+Legendre(i,prime(n))) * x^i)^2, for 0 <= k <= 2*prime(n)-2.
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0
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1, 4, 4, 1, 4, 4, 0, 0, 1, 4, 4, 0, 4, 8, 0, 0, 4, 1, 4, 8, 8, 8, 8, 8, 0, 4, 0, 0, 0, 0, 1, 4, 4, 4, 12, 12, 12, 8, 12, 12, 12, 0, 8, 8, 8, 0, 0, 0, 4, 0, 0, 1, 4, 4, 4, 12, 8, 4, 8, 4, 4, 12, 8, 12, 24, 8, 8, 8, 0, 4, 8, 4, 8, 8, 0, 4
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OFFSET
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1,2
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LINKS
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FORMULA
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For n > 1, Sum_{k=0..2*prime(n)-2} T(n,k)*x^k == (-1)^((p - 1)/2) * p mod ((x^p - 1)/(x - 1)) where p is n-th prime.
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EXAMPLE
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Triangle begins
[1, 4, 4],
[1, 4, 4, 0, 0],
[1, 4, 4, 0, 4, 8, 0, 0, 4],
[1, 4, 8, 8, 8, 8, 8, 0, 4, 0, 0, 0, 0],
[1, 4, 4, 4, 12, 12, 12, 8, 12, 12, 12, 0, 8, 8, 8, 0, 0, 0, 4, 0, 0],
[1, 4, 4, 4, 12, 8, 4, 8, 4, 4, 12, 8, 12, 24, 8, 8, 8, 0, 4, 8, 4, 8, 8, 0, 4],
...
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1 + 4*x + 4*x^2 = 4*(x^3 - 1)/(x - 1) - 3.
1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 = 4 * (x^4 - x^3 + 2*x - 1)*(x^5 - 1)/(x - 1) + 5.
1 + 4*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 8*x^6 + 4*x^8 = 4 * (x^2 - x + 2)*(x^7 - 1)/(x - 1) - 7.
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PROG
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(PARI) forprime(p=2, 30, print(Vecrev((sum(k=0, p-1, (1+kronecker(k, p))*x^k))^2, 2*p-1), ", "))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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