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
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
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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
Comments