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 A152659 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). 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). LINKS FORMULA 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). EXAMPLE 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; MAPLE 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 CROSSREFS Sequence in context: A008737 A244460 A160419 * A180214 A089452 A162487 Adjacent sequences:  A152656 A152657 A152658 * A152660 A152661 A152662 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 10 2008 STATUS approved

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