A247623 Number of paths from (0,0) to the line x = n, each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

1, 1, 2, 4, 9, 19, 44, 96, 225, 501, 1182, 2668, 6321, 14407, 34232, 78592, 187137, 432073, 1030490, 2390004, 5707449, 13286043, 31760676, 74160672, 177435297, 415382397, 994551222, 2333445468, 5590402785, 13141557519, 31500824304, 74174404608, 177880832001

Offset: 0

Author

Clark Kimberling, Sep 21 2014

Keywords

• nonn
• easy

Comments

a(n) = sum of numbers in row n of A247622.

Examples

			First 9 rows of A247622:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
3 ... 0 ... 5 ... 0 ... 1
0 ... 11 .. 0 ... 7 ... 0 ...1
11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
a(5) = 0 + 11 + 0 + 7 + 0 + 1 = 19
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2617
• Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.

Crossrefs

Cf. A247622.

Programs

Mathematica

t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1, t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = t[n - 1, 1]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}];
v = Flatten[u] (* A247622 sequence *)
TableForm[u]   (* A247622 array *)
Map[Total, u]  (* A247623 *)

Formula

Conjecture: (n+1)*a(n) +(n-3)*a(n-1) +2*(-3*n+2)*a(n-2) +2*(-3*n+8)*a(n-3) +(n-5)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Sep 23 2014