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A134426
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Triangle read by rows: T(n,k) is the number of paths of length n in the first quadrant, starting at the origin, ending at height k and consisting of 2 kind of upsteps U=(1,1) (U1 and U2), 3 kind of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1).
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1
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1, 3, 2, 11, 12, 4, 45, 62, 36, 8, 197, 312, 240, 96, 16, 903, 1570, 1440, 784, 240, 32, 4279, 7956, 8244, 5472, 2320, 576, 64, 20793, 40670, 46116, 35224, 18480, 6432, 1344, 128, 103049, 209712, 254912, 216384, 132320, 57600, 17024, 3072, 256
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OFFSET
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0,2
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COMMENTS
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T(n,0) = A001003(n+1) (the little Schroeder numbers).
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LINKS
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FORMULA
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T(n,k) = ((k+1)*2^k/(n+1))*Sum_{j=0..n-k} binomial(n+1, j)*binomial(n+1, k+j+1)*2^j (0 <= k <= n).
G.f.: g/(1-2*t*z*g), where g = 1 + 3*z*g + 2*z^2*g^2 is the g.f. of the little Schroeder numbers 1, 3, 11, 45, 197, ... (A001003).
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EXAMPLE
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T(2,1)=12 because we have 6 paths of shape FU and 6 paths of shape UF.
Triangle starts:
1;
3, 2;
11, 12, 4;
45, 62, 36, 8;
197, 312, 240, 96, 16;
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MAPLE
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T:=proc(n, k) options operator, arrow: 2^k*(k+1)*(sum(2^j*binomial(n+1, j)*binomial(n+1, k+1+j), j=0..n-k))/(n+1) end proc: for n from 0 to 8 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form
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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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