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A106509
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Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.
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4
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1, 0, 1, -1, -1, 1, 1, 0, -2, 1, 0, 1, 2, -3, 1, -1, -1, -1, 5, -4, 1, 1, 0, 0, -6, 9, -5, 1, 0, 1, 0, 6, -15, 14, -6, 1, -1, -1, 1, -6, 21, -29, 20, -7, 1, 1, 0, -2, 7, -27, 50, -49, 27, -8, 1, 0, 1, 2, -9, 34, -77, 99, -76, 35, -9, 1, -1, -1, -1, 11, -43, 111, -176, 175, -111, 44, -10, 1
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OFFSET
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0,9
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COMMENTS
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Inverse of A072405 (when this starts 1, 0, 1, ...).
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) - 2*T(n-2,k) + T(n-3,k-1) - T(n-3,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = 0, T(2,1) = T(2,0) = -1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014
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EXAMPLE
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Triangle begins:
1;
0, 1;
-1, -1, 1;
1, 0, -2, 1;
0, 1, 2, -3, 1;
-1, -1, -1, 5, -4, 1;
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
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PROG
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(Magma)
T:= func< n, k | (&+[ (-1)^j*Binomial(2*n-k-j, j): j in [0..n-k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021
(Sage)
def T(n, k): return sum( (-1)^j*binomial(2*n-k-j, j) for j in (0..n-k))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021
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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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