A151362 - OEIS

%I #40 Apr 19 2024 11:14:17

%S 1,2,18,255,4522,91896,2047452,48748986,1220457810,31779889284,

%T 854110511124,23559266827278,664125694509564,19070108145820400,

%U 556345776173277960,16455889048642607295,492658546882981692690,14907686709710614053300,455413194094843994648100

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

%H Robert Israel, <a href="/A151362/b151362.txt">Table of n, a(n) for n = 0..583</a>

%H Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/Gibbs.pdf">Composition schemes: q-enumerations and phase transitions in Gibbs models</a>, 2024. See p. 8.

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

%F a(n) = hypergeom([-n, 1/2-n], [2], 4)*binomial(2*n, n)/(n+1). - _Robert Israel_, Aug 14 2014

%F a(n) = M(2n)*C(n), where M(n) denotes Motzkin numbers, A001006, and C(n) the Catalan numbers A000108. Proof via a bijection of a pair of Dyck excursion and Motzkin excursion. - _Markus Kuba_, May 05 2022

%p seq(add(binomial(2*n,2*k)*binomial(2*k,k)/(k+1),k=0..n)*binomial(2*n,n)/(n+1),n=0..18);  # _Mark van Hoeij_, May 12 2013

%p S := proc(a) global x; series(a,x=0,20) end:

%p ogf := S(int(x^(-1/2)*int(S(x^(-1/2)*hypergeom([3/4,5/4],[2],64*x/(12*x+1)^2)/(12*x+1)^(3/2)),x),x)/(2*x)); #  _Mark van Hoeij_, Aug 14 2014

%p # third Maple program:

%p a:= proc(n) option remember; `if`(n<2, n+1,

%p      (4*n*(4*n-1)*(2*n-1)*(10*n^2-5*n-3) *a(n-1)

%p       -36*(n-1)*(2*n-1)*(4*n+1)*(-3+2*n)^2 *a(n-2))/

%p       (n*(1+2*n)*(4*n-3)*(n+1)^2))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 14 2014

%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, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}]

%Y Cf. A000108, A001006.

%K nonn,walk

%O 0,2

%A _Manuel Kauers_, Nov 18 2008