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 A098978 Triangle read by rows: T(n,k) is number of Dyck n-paths with k UUDDs, 0 <= k <= n/2. 2
 1, 1, 1, 1, 2, 3, 5, 8, 1, 13, 23, 6, 35, 69, 27, 1, 97, 212, 110, 10, 275, 662, 426, 66, 1, 794, 2091, 1602, 360, 15, 2327, 6661, 5912, 1760, 135, 1, 6905, 21359, 21534, 8022, 945, 21, 20705, 68850, 77685, 34840, 5685, 246, 1, 62642, 222892, 278192, 146092 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is the number of Łukasiewicz paths of length n having k peaks. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1). Example: T(3,1)=3 because we have HUD, UDH and U(2)DD, where H=(1,0), U(1,1), U(2)=(1,2) and D=(1,-1). (see R. P. Stanley reference). - Emeric Deutsch, Jan 06 2005 REFERENCES R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps. - Emeric Deutsch, Jan 06 2005 LINKS Alois P. Heinz, Rows n = 0..200, flattened Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8. A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. - From N. J. A. Sloane, May 05 2012 FORMULA G.f.: (1 + z^2 - t*z^2 - (-4*z + (-1 - z^2 + t*z^2)^2)^(1/2))/(2*z) = Sum_{n>=0, 0<=k<=n/2} T(n, k)z^n*t^k and it satisfies G = 1 + G^2*z + G*(-z^2 + t*z^2). T(n,k) = Sum_{j=0..floor(n/2)-k} (-1)^j * binomial(n-(j+k), j+k) * binomial(2n-3(j+k), n-(j+k)-1) * binomial(j+k, k)/(n-(j+k)). - I. Tasoulas (jtas(AT)unipi.gr), Feb 19 2006 EXAMPLE Table begins \ k  0,   1,   2, ... n 0 |  1; 1 |  1; 2 |  1,   1; 3 |  2,   3; 4 |  5,   8,   1; 5 | 13,  23,   6; 6 | 35,  69,  27,  1; 7 | 97, 212, 110, 10; 8 |275, 662, 426, 66, 1; T(3,1) = 3 because each of UUUDDD, UDUUDD, UUDDUD has one UUDD. MAPLE b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,      `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 3, 2][t])       +b(x-1, y-1, [1, 1, 4, 1][t])*`if`(t=4, z, 1))))     end: T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)): seq(T(n), n=0..15);  # Alois P. Heinz, Jun 10 2014 MATHEMATICA T[n_, k_] := Binomial[n-k, k] Binomial[2n-3k, n-k-1] HypergeometricPFQ[{k -n/2-1/2, k-n/2, k-n/2, k-n/2+1/2}, {k-2n/3, k-2n/3+1/3, k-2n/3+2/3}, 16/27]/(n-k); T[0, 0] = 1; Flatten[Table[T[n, k], {n, 0, 15}, {k, 0, n/2}]] (* Jean-François Alcover, Dec 21 2016, after 2nd formula *) CROSSREFS Column k=0 is A025242 (apart from first term). Cf. A243752. Sequence in context: A093092 A031111 A089911 * A111301 A247193 A096320 Adjacent sequences:  A098975 A098976 A098977 * A098979 A098980 A098981 KEYWORD nonn,tabf AUTHOR David Callan, Oct 24 2004 STATUS approved

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Last modified June 21 06:55 EDT 2021. Contains 345358 sequences. (Running on oeis4.)