cp's OEIS Frontend

A129637 Number of n-step paths that can go {west, southeast, southwest, northwest} on a 240-degree wedge on the equilateral triangular lattice.

%I A129637 #30 Aug 06 2024 04:49:22
%S A129637 1,3,11,41,157,607,2367,9277,36505,144059,569779,2257521,8957109,
%T A129637 35579351,141460391,562871557,2241129905,8928207987,35584894299,
%U A129637 141886838329,565938926669
%N A129637 Number of n-step paths that can go {west, southeast, southwest, northwest} on a 240-degree wedge on the equilateral triangular lattice.
%C A129637 If we use the "hour hand" positions at 1, 3, 5, 7, 9, and 11 o'clock on a 12-hour clock to specify directions in the triangular lattice, the allowable steps are in directions 5, 7, 9, and 11 and the path is restricted to stay on or above the 1-7 line. In the Mathematica recurrence below, a(n,k) denotes the number of paths of length n ending k units from the 1-7 line, counted by the last step. - _David Callan_, Jul 22 2008
%H A129637 Andrew Howroyd, <a href="/A129637/b129637.txt">Table of n, a(n) for n = 0..200</a>
%H A129637 A. Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">Computer Algebra for Lattice Path Combinatorics</a>, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H A129637 Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%F A129637 Recurrence: (-28-28*n)*a(n) + (-13-n)*a(1+n) + (21+6*n)*a(n+2) + (-4-n)*a(n+3), a(0) = 1, a(1) = 3, a(2) = 11.
%F A129637 G.f.: (1/4*i)*sqrt(-1+2*t+7*t^2)/((-1+4*t)*t)-(1/4)*(-1+5*t)/(t*(-1+4*t)), where i is the imaginary unit.
%F A129637 Differential equation: -(1+28*t^3-6*t+t^2)*t*((d/dt)f(t)) + (9*t-12*t^2-1-28*t^3)*f(t) + 1 - 3*t, f(0) = 1.
%F A129637 a(n) = Sum_{k=0..n+1}(binomial(n+1,k)*Sum_{i=0..n}(2^(i+k-1)*binomial(k,i)*(-1)^(n-i)*binomial(n-i-1,k-i-1)))/(n+1). - _Vladimir Kruchinin_, Feb 28 2016
%F A129637 a(n) ~ 2^(2*n-1). - _Vaclav Kotesovec_, Feb 28 2016
%e A129637 a(1) = 3 because only 3 out of the 4 steps are permissible from the origin;
%e A129637 a(2) = 11 because the northwest and west steps are followed by 4 permissible steps each, but the southwest step is only followed by 3 permissible steps.
%t A129637 a[0,0]=1; a[n_,k_]/;k<0 || k>n := 0; a[n_,k_]/;0<=k<=n := a[n,k] = 2a[n-1,k-1] + a[n-1,k] + a[n-1,k+1]; a[n_]:=Sum[a[n,k],{k,0,n}]; Table[a[n],{n,0,10}] (* _David Callan_, Jul 22 2008 *)
%o A129637 (Maxima)
%o A129637 a(n):=sum(binomial(n+1,k)*sum(2^(i+k-1)*binomial(k,i)*(-1)^(n-i)*binomial(n-i-1,k-i-1),i,0,n),k,0,n+1)/(n+1); /* _Vladimir Kruchinin_, Feb 28 2016 */
%o A129637 (PARI) a(n) = {sum(k=0, n+1, binomial(n+1, k) * sum(i=0,n, 2^(i+k-1)*binomial(k,i)*(-1)^(n-i)*binomial(n-i-1,k-i-1)))/(n+1)} \\ _Andrew Howroyd_, Dec 22 2017
%Y A129637 Cf. A129400.
%K A129637 nonn
%O A129637 0,2
%A A129637 Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 31 2007