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%I #54 Sep 08 2022 08:44:59
%S 1,5,36,286,2380,20349,177100,1560780,13884156,124403620,1121099408,
%T 10150595910,92263734836,841392966470,7694644696200,70539987842520,
%U 648045936942300,5964720367660956,54991682779774384,507749884105448600,4694436188839116720
%N a(n) = (4n+1)*binomial(4n,n)/(3n+1).
%C Central terms of the triangles in A122366 and A111418. - _Reinhard Zumkeller_, Aug 30 2006 and Mar 14 2014
%C a(n) is the number of paths from (0,0) to (4n,n), taking north and east steps while avoiding exactly 2 consecutive north steps. - _Shanzhen Gao_, Apr 15 2010
%H Reinhard Zumkeller, <a href="/A052203/b052203.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = C(4n+1, n); a(n) is asymptotic to c/sqrt(n)*(256/27)^n with c=0.614... - _Benoit Cloitre_, Jan 27 2003 [c = 2^(5/2)/(3^(3/2)*sqrt(Pi)) = 0.61421182128... - _Vaclav Kotesovec_, Feb 14 2019]
%F G.f.: g^2/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Nov 11 2011
%F G.f.: hypergeom([1/2, 3/4, 5/4], [2/3, 4/3], (256/27)*x). - _Robert Israel_, Aug 07 2014
%F D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Nov 26 2012
%F From _Peter Bala_, Nov 04 2015: (Start)
%F The o.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293.
%F More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A257633 (k = 2), A224274 (k = 3) and A004331 (k = 4). (End)
%F a(n) = [x^n] 1/(1 - x)^(3*n+2). - _Ilya Gutkovskiy_, Oct 03 2017
%t Table[Binomial[4 n + 1, n], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 07 2014 *)
%o (Haskell)
%o a052203 n = a122366 (2 * n) n -- _Reinhard Zumkeller_, Mar 14 2014
%o (Magma) [Binomial(4*n+1, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%o (PARI) vector(30, n, n--; (4*n+1)*binomial(4*n,n)/(3*n+1)) \\ _Altug Alkan_, Nov 05 2015
%Y Cf. A002293, A051944, A004331, A005810, A224274, A257633, A262977.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jan 28 2000
%E More terms from _James A. Sellers_, Jan 31 2000
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