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A151332 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 4 n steps taken from {(-1, -1), (-1, 1), (1, 0)}.

%I A151332 #27 Dec 27 2023 01:20:09
%S A151332 1,2,28,660,20020,705432,27457584,1147334760,50561468100,
%T A151332 2322279359400,110250966574320,5377893986141040,268315541493159888,
%U A151332 13645106597301720800,705378072079232798400,36985702814877062972880,1963555139681260758978660,105393959626252993455319560
%N A151332 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 4 n steps taken from {(-1, -1), (-1, 1), (1, 0)}.
%C A151332 a(n) is also the number of words of 4n length consisting of 2n X's, n Y's and n Z's such that any initial segment of the string has at least as many X's as Y+Z's, and at least as many Y's as Z's. - _Istvan Marosi_, Apr 27 2014
%H A151332 Alois P. Heinz, <a href="/A151332/b151332.txt">Table of n, a(n) for n = 0..558</a>
%H A151332 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.
%F A151332 a(n) = A000108(n)*A000108(2n). - _Istvan Marosi_, Apr 27 2014
%F A151332 a(n) = A056040(4*n)*A056040(2*n)/A000384(n+1). - _Peter Luschny_, Apr 28 2014
%F A151332 G.f.: hypergeom([1/4, 1/2, 3/4], [3/2, 2], 64*x). - _Robert Israel_, Aug 14 2014
%F A151332 D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -4*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%p A151332 a:= proc(n) option remember; `if`(n=0, 1,
%p A151332       (4*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1))/(n*(n+1)*(2*n+1)))
%p A151332     end:
%p A151332 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 27 2014
%p A151332 S := proc(a) global x; series(a,x=0,20) end:
%p A151332 ogf := S(int(S(x^(-1/2)*hypergeom([1/4,3/4],[2],64*x)),x)/(2*x^(1/2)));  # _Mark van Hoeij_, Aug 14 2014
%t A151332 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[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 4 n], {n, 0, 25}]
%K A151332 nonn,walk
%O A151332 0,2
%A A151332 _Manuel Kauers_, Nov 18 2008