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A117438
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Triangle T(n, k) = binomial(2*n-k, k)*(-4)^(n-k), read by rows.
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3
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1, -4, 1, 16, -12, 1, -64, 80, -24, 1, 256, -448, 240, -40, 1, -1024, 2304, -1792, 560, -60, 1, 4096, -11264, 11520, -5376, 1120, -84, 1, -16384, 53248, -67584, 42240, -13440, 2016, -112, 1, 65536, -245760, 372736, -292864, 126720, -29568, 3360, -144, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n, k) = binomial(2*n-k, k)*(-4)^(n-k).
Sum_{k=0..n} T(n, k) = (-1)^n*(2*n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A117439(n) (upward diagonal sums).
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EXAMPLE
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Triangle begins
1;
-4, 1;
16, -12, 1;
-64, 80, -24, 1;
256, -448, 240, -40, 1;
-1024, 2304, -1792, 560, -60, 1;
4096, -11264, 11520, -5376, 1120, -84, 1;
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MATHEMATICA
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Table[Binomial[2*n-k, k]*(-4)^(n-k), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
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PROG
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(Sage) flatten([[binomial(2*n-k, k)*(-4)^(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 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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