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 A291883 Number T(n,k) of symmetrically unique Dyck paths of semilength n and height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 12
 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 9, 11, 4, 1, 0, 1, 19, 31, 19, 5, 1, 0, 1, 35, 91, 69, 29, 6, 1, 0, 1, 71, 250, 252, 127, 41, 7, 1, 0, 1, 135, 690, 855, 540, 209, 55, 8, 1, 0, 1, 271, 1863, 2867, 2117, 1005, 319, 71, 9, 1, 0, 1, 527, 5017, 9339, 8063, 4411, 1705, 461, 89, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = (A080936(n,k) + A132890(n,k))/2. Sum_{k=1..n} k * T(n,k) = A291886(n). EXAMPLE : T(4,2) = 5: /\ /\ /\/\ /\ /\ /\/\/\ : /\/\/ \ /\/ \/\ /\/ \ / \/ \ / \ : Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 1, 2, 1; 0, 1, 5, 3, 1; 0, 1, 9, 11, 4, 1; 0, 1, 19, 31, 19, 5, 1; 0, 1, 35, 91, 69, 29, 6, 1; 0, 1, 71, 250, 252, 127, 41, 7, 1; 0, 1, 135, 690, 855, 540, 209, 55, 8, 1; MAPLE b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y0, b(x-1, y-1, k), 0)) end: g:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0, g(x-2, y-1, k), 0)+ g(x-2, y+1, max(y+1, k))) end: T:= n-> (p-> seq(coeff(p, z, i)/2, i=0..n))(b(2*n, 0\$2)+g(2*n, 0\$2)): seq(T(n), n=0..14); MATHEMATICA b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y < x - 1, b[x - 1, y + 1, Max[y + 1, k]], 0] + If[y > 0, b[x - 1, y - 1, k], 0]]; g[x_, y_, k_] := g[x, y, k] = If[x == 0, z^k, If[y > 0, g[x - 2, y - 1, k], 0] + g[x - 2, y + 1, Max[y + 1, k]]]; T[n_] := Function[p, Table[Coefficient[p, z, i]/2, {i, 0, n}]][b[2*n, 0, 0] + g[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *) PROG (Python) from sympy.core.cache import cacheit from sympy import Poly, Symbol, flatten z=Symbol('z') @cacheit def b(x, y, k): return z**k if x==0 else (b(x - 1, y + 1, max(y + 1, k)) if y0 else 0) @cacheit def g(x, y, k): return z**k if x==0 else (g(x - 2, y - 1, k) if y>0 else 0) + g(x - 2, y + 1, max(y + 1, k)) def T(n): return 1 if n==0 else [i//2 for i in Poly(b(2*n, 0, 0) + g(2*n, 0, 0)).all_coeffs()[::-1]] print(flatten(map(T, range(15)))) # Indranil Ghosh, Sep 06 2017 CROSSREFS Columns k=0-10 give: A000007, A057427, A056326, A291887, A291888, A291889, A291890, A291891, A291892, A291893, A291894. Main and first two lower diagonals give A000012, A001477, A028387(n-1) for n>0. Row sums give A007123(n+1). T(2n,n) give A291885. Cf. A080936, A132890, A291886. Sequence in context: A370773 A119331 A351641 * A361957 A239145 A327127 Adjacent sequences: A291880 A291881 A291882 * A291884 A291885 A291886 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 05 2017 STATUS approved

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Last modified April 14 18:28 EDT 2024. Contains 371667 sequences. (Running on oeis4.)