A151366 - OEIS

%I #30 Dec 27 2023 21:33:58

%S 1,0,2,2,12,30,130,462,1946,7980,34776,153120,694056,3194334,14971242,

%T 71133062,342500730,1667918824,8208038124,40772105244,204270936480,

%U 1031413134960,5245260798960,26850869456400,138289429433200,716247599547360,3729128330979200,19510354349803200,102540704879774160

%N Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0)}.

%H Michael De Vlieger, <a href="/A151366/b151366.txt">Table of n, a(n) for n = 0..500</a>

%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, <a href="https://arxiv.org/abs/1911.10288">On sequences associated to the invariant theory of rank two simple Lie algebras</a>, arXiv:1911.10288 [math.CO], 2019.

%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, <a href="https://arxiv.org/abs/2110.13753">On some combinatorial sequences associated to invariant theory</a>, arXiv:2110.13753 [math.CO], 2021.

%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%F G.f.: ((2*x+1)*(hypergeom([-2/3, -1/3],[1],27*x^2*(2*x+1))+4*x*hypergeom([-1/3, 1/3],[2],27*x^2*(2*x+1)))/(3*x+1)-1-3*x-5*x^2)/(3*x^3). - _Mark van Hoeij_, Aug 17 2014

%F Conjecture: (n+4)*(n+3)*a(n) -n*(n-1)*a(n-1) -12*(2*n+1)*(n-1)*a(n-2) -36*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jul 21 2017

%F G.f.: (-5*x^2-3*x-1)/(3*x^3)+(1+3*x)^3*hypergeom([1/4, 3/4],[2],64*x*(1+3*x)^3/(1+24*x+36*x^2)^2)/(3*x^3*(1+24*x+36*x^2)^(1/2)). - _Mark van Hoeij_, Jul 27 2021

%t aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008