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 A118920 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross the x-axis k times (n>=1, k>=0). 2
 2, 4, 2, 10, 8, 2, 28, 28, 12, 2, 84, 96, 54, 16, 2, 264, 330, 220, 88, 20, 2, 858, 1144, 858, 416, 130, 24, 2, 2860, 4004, 3276, 1820, 700, 180, 28, 2, 9724, 14144, 12376, 7616, 3400, 1088, 238, 32, 2, 33592, 50388, 46512, 31008, 15504, 5814, 1596, 304, 36, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). Row sums are the central binomial coefficients (A000984). T(n,0)=2*A000108(n) (the Catalan numbers doubled). T(n,1)=2*A002057(n-2). Sum(k*T(n,k),k>=0)=2*A008549(n-1). For crossings of the x-axis in one direction, see A118919. This triangle is related to paired pattern P_3 and P_4 defined in the Pan & Remmel link. - Ran Pan, Feb 01 2016 LINKS Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016. FORMULA T(n,k) = 2*(k+1)*binomial(2*n,n-k-1)/n. G.f.: G(t,z)=2*z*C^2/(1-t*z*C^2), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. More generally, the trivariate g.f. G=G(x,y,z), where x (y) marks number of downward (upward) crossings of the x-axis, is given by G = z*C^2*(2+(x+y)*z*C^2)/(1-x*y*z^2*C^4). EXAMPLE T(3,1)=8 because we have ud|dudu,ud|dduu,udud|du,uudd|du,du|udud,du|uudd, dudu|ud and dduu|ud (the crossings of the x-axis are shown by |). Triangle starts: 2; 4,2; 10,8,2; 28,28,12,2; 84,96,54,16,2; MAPLE T:=(n, k)->2*(k+1)*binomial(2*n, n-k-1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form MATHEMATICA Table[2 (k + 1) Binomial[2 n, n - k - 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Feb 01 2016 *) PROG (Sage) # Algorithm of L. Seidel (1877) # Prints the first n rows of the triangle. def A118920_triangle(n) :     D = *(n+2); D = 2     b = True; h = 1     for i in range(2*n) :         if b :             for k in range(h, 0, -1) : D[k] += D[k-1]             h += 1         else :             for k in range(1, h, 1) : D[k] += D[k+1]         b = not b         if b : print([D[z] for z in (1..h-1)]) A118920_triangle(10)  # Peter Luschny, Oct 19 2012 (PARI) T(n, k) = 2*(k+1)*binomial(2*n, n-k-1)/n \\ Charles R Greathouse IV, Feb 01 2016 CROSSREFS Cf. A000984, A000108, A002057, A008549, A118919. Sequence in context: A236959 A097577 A097692 * A305260 A162982 A259707 Adjacent sequences:  A118917 A118918 A118919 * A118921 A118922 A118923 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, May 06 2006 STATUS approved

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Last modified October 25 19:28 EDT 2020. Contains 338012 sequences. (Running on oeis4.)