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 A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). 5
 1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920. Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006 Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008 LINKS FORMULA T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=A039599(n,2k)+A039599(n,2k+1). - Philippe Deléham, Nov 11 2008 EXAMPLE T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \). Triangle starts: 1; 2; 5,1; 14,6; 42,27,1; 132,110,10; MAPLE T:=(n, k)->(2*k+1)*binomial(2*n+2, n-2*k)/(n+1): for n from 0 to 13 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form PROG (PARI) T(n, k)=if(n<2*k || k<0, 0, (2*k+1)*binomial(2*n+2, n-2*k)/(n+1)) - Paul D. Hanna, May 10 2006 CROSSREFS Cf. A000984, A000108, A003517, A003519, A008549, A118920. Cf. A119243 (eigenvector), A119245 (variant). Sequence in context: A263771 A073187 A138159 * A319120 A274404 A101282 Adjacent sequences:  A118916 A118917 A118918 * A118920 A118921 A118922 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 06 2006 STATUS approved

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Last modified October 20 17:33 EDT 2018. Contains 316393 sequences. (Running on oeis4.)