This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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