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A248975
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Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+(-1)^k)^k for 0 <= k <= n.
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2
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1, 2, 1, -1, -1, 1, -12, -10, 4, 1, 45, 34, -14, -3, 1, 406, 319, -124, -33, 6, 1, -2357, -1847, 731, 187, -39, -5, 1, -26968, -21188, 8312, 2182, -424, -68, 8, 1, 223769, 175700, -69052, -18034, 3566, 548, -76, -7, 1, 3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1
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OFFSET
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0,2
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COMMENTS
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Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+1)^0 + A_1*(x-1)^1 + A_2*(x+1)^2 + A_3*(x-1)^3 + ... + A_n*(x+(-1)^n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
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LINKS
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FORMULA
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T(n,n-1) = 1 - n*(-1)^n for n > 0.
T(n,n-2) = (1-n)*((3/2)*n-(-1)^n) + 1 for n > 1.
T(n,0) = 1 - sum(i=1..n) (-1)^i*T(n,i) = 1 + T(n,1) - T(n-2) + T(n-3) - ... + (-1)^(n-1)*T(n,n-1) + (-1)^n.
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EXAMPLE
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1;
2, 1;
-1, -1, 1;
-12, -10, 4, 1;
45, 34, -14, -3, 1;
406, 319, -124, -33, 6, 1;
-2357, -1847, 731, 187, -39, -5, 1;
-26968, -21188, 8312, 2182, -424, -68, 8, 1;
223769, 175700, -69052, -18034, 3566, 548, -76, -7, 1;
3096810, 2432333, -955048, -250126, 49052, 7730, -1000, -115, 10, 1
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PROG
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(PARI) a(n, j) = if(j==n, return(1)); if(j!=n, return(1-sum(i=1, n-j, (-1)^(i*(j+1))*binomial(i+j, i)*a(n, i+j))))
for(n=0, 15, for(j=0, n, print1(a(n, j), ", ")))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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