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Search: id:A152659
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| A152659 |
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Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) and having k turns (NE or EN) (1<=k<=2n-1). |
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+0
1
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| 2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 6, 18, 18, 18, 6, 2, 2, 8, 32, 48, 72, 48, 32, 8, 2, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2, 2, 12, 72, 180, 450, 600, 800, 600, 450, 180, 72, 12, 2, 2, 14, 98, 294, 882, 1470, 2450, 2450, 2450, 1470, 882, 294, 98, 14, 2, 2, 16, 128, 448
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OFFSET
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1,1
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COMMENT
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Row n has 2n-1 entries.
Sum of entries of row n = binom(2n,n) = A000984(n) (the central binomial coefficients).
Sum(k*T(n,k),k=0..2n-1)=n*binom(2n,n) = A005430(n).
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FORMULA
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T(n,2k)=2*binom(n-1,k-1)binom(n-1,k);
T(n,2k-1)=2[binom(n-1,k-1)]^2.
G.f. = [1+t*r(t^2,z)]/[1-t*r(t^2,z)], where r(t,z) is the Narayana function, defined by r=z(1+r)(1+tr).
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EXAMPLE
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T(3,2)=4 because we have ENNNEE, EENNNE, NEEENN and NNEEEN.
Triangle starts:
2;
2,2,2;
2,4,8,4,2;
2,6,18,18,18,6,2;
2,8,32,48,72,48,32,8,2;
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MAPLE
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T := proc (n, k) if `mod`(k, 2) = 0 then 2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: for n to 9 do seq(T(n, k), k = 1 .. 2*n-1) end do; # yields sequence in triangular form
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CROSSREFS
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A000984, A005430
Sequence in context: A029103 A008737 A160419 this_sequence A089452 A162487 A115101
Adjacent sequences: A152656 A152657 A152658 this_sequence A152660 A152661 A152662
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2008
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