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A151341 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, 0)}.

%I A151341 #24 Sep 08 2022 08:45:38
%S A151341 1,1,4,20,126,882,6732,54483,461890,4059770,36749648,340841228,
%T A151341 3226474132,31079221500,303907314960,3010947401340,30176604541890,
%U A151341 305544118531410,3122029001183400,32162924697555960,333798776316127980,3487606641543204180,36662847052669011720,387575171486963664750
%N A151341 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, 0)}.
%H A151341 G. C. Greubel, <a href="/A151341/b151341.txt">Table of n, a(n) for n = 0..925</a>
%H A151341 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 A151341 a(n) = A000108(n)*A001006(n).
%F A151341 Conjecture: n*(n+2)*(n+1)*a(n) - 2*n*(2*n-1)*(2*n+1)*a(n-1) -12*(n-1)*(2*n-1)*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Jul 21 2017
%p A151341 ogf := subs(t=sqrt(x),series( Int(Int(2*hypergeom([3/2, 3/2],[3],16*t^2/(1+4*t^2))/(1+4*t^2)^(3/2),t),t)/t^2, t=0, 60)); # _Mark van Hoeij_, Aug 17 2014
%t A151341 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, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}]
%t A151341 Table[CatalanNumber[n]*(3/2)^(n+2)*Sum[CatalanNumber[k-1]*Binomial[k, n+2 -k]/3^k, {k, 1, n+2}], {n, 0, 25}] (* _G. C. Greubel_, Mar 11 2019 *)
%o A151341 (PARI) {a(n) = (3/2)^(n+2)*(binomial(2*n, n)/(n+1))*sum(k=1, n+2, binomial(k, n-k+2)*binomial(2*k-2, k-1)/(3^k*k))};
%o A151341 vector(25, n, n--; a(n)) \\ _G. C. Greubel_, Mar 11 2019
%o A151341 (Magma) [(3/2)^(n+2)*Catalan(n)*(&+[Binomial(k, n-k+2)*Catalan(k-1)/(3^k): k in [1..n+2]]): n in [0..25]]; // _G. C. Greubel_, Mar 11 2019
%o A151341 (Sage) [(3/2)^(n+2)*catalan_number(n)*sum(binomial(k, n-k+2)* catalan_number(k-1)/3^k for k in (1..n+2)) for n in (0..25)] # _G. C. Greubel_, Mar 11 2019
%Y A151341 Cf. A000108, A001006.
%K A151341 nonn,walk
%O A151341 0,3
%A A151341 _Manuel Kauers_, Nov 18 2008