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A111911 a(n) = (4*n+1)!/( (2*n+1)! * ((n+1)!)^2 ).

%I A111911 #23 Feb 12 2021 01:34:08
%S A111911 1,5,84,2145,68068,2469012,98062800,4159088505,185392049700,
%T A111911 8592433629780,410935420867920,20167102448028900,1011343194858833424,
%U A111911 51656474975499371600,2680436673901084633920,141007991981718802584105,7507710828193055843153700
%N A111911 a(n) = (4*n+1)!/( (2*n+1)! * ((n+1)!)^2 ).
%C A111911 Main diagonal of the square array A111910, i.e., a(n) = A111910(n,n).
%H A111911 G. C. Greubel, <a href="/A111911/b111911.txt">Table of n, a(n) for n = 0..550</a>
%H A111911 G. Kreweras and H. Niederhausen, <a href="http://dx.doi.org/10.1016/S0195-6698(81)80020-0">Solution of an enumerative problem connected with lattice paths</a>, European J. Combin., 2 (1981), 55-60.
%F A111911 G.f.: expression with a 2F1 function and an anti-derivative, see Maple program below. - _Mark van Hoeij_, May 01 2013
%F A111911 a(n) ~ 2^(6*n + 1/2) / (Pi * n^3). - _Vaclav Kotesovec_, Dec 16 2017
%F A111911 D-finite with recurrence (2*n+1)*(n+1)^2*a(n) -4*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021
%F A111911 From _G. C. Greubel_, Feb 12 2021: (Start)
%F A111911 a(n) = binomial(4*n+1, 2*n+1)*binomial(2*n, n)/(n+1)^2.
%F A111911 a(n) = ((4*n+1)/(n+1))*C_{n}*C_{2*n}, where C_{n} are the Catalan numbers (A000108). (End)
%p A111911 a:=n->1/(2*n+1)!*(4*n+1)!/(n+1)!^2: seq(a(n),n=1..17);
%p A111911 ogf := -1/(4*x)-Int(x^(-3/2)*hypergeom([-1/4, 1/4],[1],64*x),x)/(8*x^(1/2));
%p A111911 series( eval(ogf, Int = proc(a,x) int(series(a,x=0,32),x) end), x=0, 30); # _Mark van Hoeij_, May 01 2013
%t A111911 Table[((4*n+1)/(n+1))*CatalanNumber[n]*CatalanNumber[2*n], {n,0,30}] (* _G. C. Greubel_, Feb 12 2021 *)
%o A111911 (Sage) [((4*n+1)/(n+1))*catalan_number(n)*catalan_number(2*n) for n in (0..30)] # _G. C. Greubel_, Feb 12 2021
%o A111911 (Magma) [((4*n+1)/(n+1))*Catalan(n)*Catalan(2*n): n in [0..30]]; // _G. C. Greubel_, Feb 12 2021
%Y A111911 Cf. A111910.
%Y A111911 Cf. A000108.
%K A111911 nonn
%O A111911 0,2
%A A111911 _Emeric Deutsch_, Aug 19 2005