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Search: id:A134426
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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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+0
2
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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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COMMENT
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T(n,0)=A001003(n+1) (the little Schroder numbers). Row sums yield A134425.
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FORMULA
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T(n,k)=[(k+1)2^k/(n+1)]Sum(binom(n+1,j)binom(n+1,k+j+1)2^j, j=0..n-k) (0<=k<=n). G.f.=g/(1-2tzg), where g=1+3zg+2z^2 g^2 is the g.f. of the little Schroder 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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Cf. A001003, A134425.
Sequence in context: A072634 A086194 A074246 this_sequence A122672 A052973 A087956
Adjacent sequences: A134423 A134424 A134425 this_sequence A134427 A134428 A134429
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 05 2007
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