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A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.

%I A151403 #88 Oct 04 2024 08:53:19
%S A151403 1,4,32,320,3584,43008,540672,7028736,93716480,1274544128,17611882496,
%T A151403 246566354944,3489862254592,49855175065600,717914520944640,
%U A151403 10409760553697280,151860036312760320,2227280532587151360,32823081532863283200,485781606686376591360,7217326727911880785920
%N A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.
%C A151403 Essentially the same as A052704. - _R. J. Mathar_, Nov 27 2008
%C A151403 From _Joerg Arndt_, Oct 22 2012: (Start)
%C A151403 Number of strings of length 2*n of four different types of balanced parentheses.
%C A151403 The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
%C A151403 Number of Dyck paths of length 2n in which the step U=(1,1) comes in 4 colors. - _José Luis Ramírez Ramírez_, Jan 31 2013
%D A151403 Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 106.
%H A151403 Vincenzo Librandi, <a href="/A151403/b151403.txt">Table of n, a(n) for n = 0..200</a>
%H A151403 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], 2008-2009.
%H A151403 Alon Regev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Regev/alon3.html">Enumerating Triangulations by Parallel Diagonals</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.5; <a href="http://arxiv.org/abs/1208.3915">arXiv preprint</a>, arXiv:1208.3915 [math.CO], 2012.
%F A151403 a(n) = 4^n*A000108(n). - _Philippe Deléham_, Feb 01 2009
%F A151403 a(n) = Integral_{x=-2..2} (2*x)^(2*n)*sqrt((2-x)*(2+x))/(2*Pi) dx. - _Peter Luschny_, Sep 11 2011
%F A151403 E.g.f.: KummerM(1/2, 2, 16*x). - _Peter Luschny_, Aug 26 2012
%F A151403 G.f.: 2/(1 + sqrt(1-16*x)) = 1/U(0) where U(k) = 1 - 4*x/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 30 2012
%F A151403 G.f.: c(4*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013
%F A151403 a(n) = sum{k=0..n} A085880(n,k)*3^k. - _Philippe Deléham_, Nov 15 2013
%F A151403 a(n) = 4^n*hypergeom([1-n,-n],[2],1). - _Peter Luschny_, Sep 22 2014
%F A151403 a(n) = 4^n*(2*n)!*[x^(2*n)]hypergeom([],[2],x^2). - _Peter Luschny_, Jan 31 2015
%F A151403 a(n) ~ 2^(4*n+2)/((2*n+1)*sqrt(Pi*(4*n+5))). - _Peter Luschny_, Jan 31 2015
%F A151403 D-finite with recurrence: (n+1)*a(n) +8*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 21 2020
%F A151403 Sum_{n>=0} 1/a(n) = 88/75 + 128*arctan(1/sqrt(15)) / (75*sqrt(15)). - _Vaclav Kotesovec_, Nov 23 2021
%F A151403 Sum_{n>=0} (-1)^n/a(n) = 248/289 - 384*arctanh(1/sqrt(17)) / (289*sqrt(17)). - _Amiram Eldar_, Jan 25 2022
%F A151403 G.f.: 1/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-...)))))))))(continued fraction), cf. g.f. by _Sergei N. Gladkovskii_. - _Nikolaos Pantelidis_, Nov 21 2022
%F A151403 a(n) = 4*A269796(n-1) for n>0. - _Hugo Pfoertner_, Oct 04 2024
%p A151403 A151403_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A151403 for w from 1 to n do a[w] := 4*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od; convert(a,list) end: A151403_list(20); # _Peter Luschny_, May 19 2011
%p A151403 seq(4^n*(2*n)!*coeff(series(hypergeom([],[2],x^2),x,2*n+2),x,2*n),n=0..20); # _Peter Luschny_, Jan 31 2015
%t A151403 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, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
%o A151403 (Magma) [4^n * Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Oct 24 2012
%o A151403 (Sage)
%o A151403 A151403 = lambda n: 4^n*hypergeometric([1-n,-n],[2],1)
%o A151403 [Integer(A151403(n).n()) for n in range(21)] # _Peter Luschny_, Sep 22 2014
%Y A151403 Cf. A000108, A052704, A085880, A269796.
%K A151403 nonn,walk
%O A151403 0,2
%A A151403 _Manuel Kauers_, Nov 18 2008