cp's OEIS Frontend

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.

%I A247623 #13 May 11 2018 20:42:45
%S A247623 1,1,2,4,9,19,44,96,225,501,1182,2668,6321,14407,34232,78592,187137,
%T A247623 432073,1030490,2390004,5707449,13286043,31760676,74160672,177435297,
%U A247623 415382397,994551222,2333445468,5590402785,13141557519,31500824304,74174404608,177880832001
%N 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.
%C A247623 a(n) = sum of numbers in row n of A247622.
%H A247623 Michael De Vlieger, <a href="/A247623/b247623.txt">Table of n, a(n) for n = 0..2617</a>
%H A247623 Axel Bacher, <a href="https://arxiv.org/abs/1802.06030">Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths</a>, arXiv:1802.06030 [cs.DS], 2018.
%F A247623 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
%e A247623 First 9 rows of A247622:
%e A247623 1
%e A247623 0 ... 1
%e A247623 1 ... 0 ... 1
%e A247623 0 ... 3 ... 0 ... 1
%e A247623 3 ... 0 ... 5 ... 0 ... 1
%e A247623 0 ... 11 .. 0 ... 7 ... 0 ...1
%e A247623 11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
%e A247623 0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
%e A247623 45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
%e A247623 a(5) = 0 + 11 + 0 + 7 + 0 + 1 = 19
%t A247623 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}];
%t A247623 v = Flatten[u] (* A247622 sequence *)
%t A247623 TableForm[u]   (* A247622 array *)
%t A247623 Map[Total, u]  (* A247623 *)
%Y A247623 Cf. A247622.
%K A247623 nonn,easy
%O A247623 0,3
%A A247623 _Clark Kimberling_, Sep 21 2014