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A300953
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Number T(n,k) of Dyck paths of semilength n such that 2*k is the difference between the area above the path and the area below the path, measured within the smallest enclosing rectangle based on the x-axis; triangle T(n,k), n>=0, -floor((n-1)^2/4) <= k <= floor((n-1)^2/4), read by rows.
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4
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1, 1, 2, 1, 2, 2, 2, 0, 7, 0, 5, 1, 2, 3, 6, 7, 8, 6, 6, 3, 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7, 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4, 2, 0, 11, 0, 29, 0, 63, 0, 115, 0, 176, 0, 238, 0, 255, 0, 230, 0, 169, 0, 92, 0, 41, 0, 9
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OFFSET
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0,3
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LINKS
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FORMULA
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Sum_{k = -floor((n-1)^2/4)..floor((n-1)^2/4)} k * T(n,k) = A300996(n).
T(n,-floor((n-1)^2/4)) = A040001(n).
T(n, floor((n-1)^2/4)) = A026741(n+1) for n > 2.
T(n,k) = 0 iff n is even and k is odd or abs(k) > floor(n*(n-1)/6).
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EXAMPLE
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.______.
| /\/\ | , rectangle area: 12, above path area: 5,
T(3,-1) = 1: |/____\| , below path area: 7, difference: (5-7) = 2 * (-1).
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/\
/ \
T(3,0) = 2: / \ /\/\/\ .
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/\ /\
T(3,1) = 2: / \/\ /\/ \ .
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Triangle T(n,k) begins:
: 1 ;
: 1 ;
: 2 ;
: 1, 2, 2 ;
: 2, 0, 7, 0, 5 ;
: 1, 2, 3, 6, 7, 8, 6, 6, 3 ;
: 2, 0, 9, 0, 20, 0, 35, 0, 34, 0, 25, 0, 7 ;
: 1, 2, 4, 8, 10, 17, 23, 30, 38, 43, 46, 48, 42, 41, 26, 26, 12, 8, 4 ;
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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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