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A265613 a(n) = CatalanNumber(n+1)*n*(3*n^2+5*n+2)/((4+n)*(3+n)).

%I A265613 #29 Oct 09 2017 14:36:42
%S A265613 0,1,8,44,210,935,4004,16744,68952,281010,1136960,4576264,18349630,
%T A265613 73370115,292746300,1166182800,4639918800,18443677230,73261092240,
%U A265613 290845019400,1154169552900,4578702310182,18159992594568,72014135814704,285542883894800,1132125641947300
%N A265613 a(n) = CatalanNumber(n+1)*n*(3*n^2+5*n+2)/((4+n)*(3+n)).
%C A265613 This is row n=8 in the array A(n,k) = (rf(k+n-2,k-1)-(k-1)*(k-2)*rf(k+n-2, k-3))/ (k-1)! if n>=3 and A(n,0)=0, A(n,1)=1, A(n,2)=n; rf(n,k) denotes the rising factorial. See the cross-references for other values of n and the table in A264357.
%H A265613 Robert Israel, <a href="/A265613/b265613.txt">Table of n, a(n) for n = 0..1630</a>
%F A265613 G.f.: I*(14*x^2+I*sqrt(4*x-1)*(4*x^2-7*x+2)-11*x+2*(1-x^3))/(2*x^4*sqrt(4*x-1)).
%F A265613 a(n) = (4^(n+1)*n*(n+1)*(3*n+2)*Gamma(n+3/2))/(sqrt(Pi)*Gamma(n+5)).
%F A265613 a(n) = (rf(n+6, n-1)-(n-1)*(n-2)*rf(n+6, n-3))/(n-1)! for n>=3, rf(n,k) the rising factorial.
%F A265613 a(n) = a(n-1)*((2*(n+1))*(3*n+2)*(1+2*n)/((n-1)*(3*n-1)*(4+n))) for n>=2.
%F A265613 a(n) ~ 4^n*(12-(191/2)/n+(17595/32)/n^2-(705005/256)/n^3+(104705937/8192)/ n^4-...)/sqrt(n*Pi).
%F A265613 a(n) = [x^n] x*(1 + x)/(1 - x)^(n+5). - _Ilya Gutkovskiy_, Oct 09 2017
%p A265613 A265613 := n -> (4*4^n*n*(n+1)*(3*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*GAMMA(n+5)):
%p A265613 seq(simplify(A265613(n)), n=0..25);
%t A265613 Table[SeriesCoefficient[I (14 x^2 + I Sqrt[4 x - 1] (4 x^2 - 7 x + 2) - 11 x + 2 (1 - x^3))/(2 x^4 Sqrt[4 x - 1]), {x, 0, n}], {n, 0, 25}]
%t A265613 (* or *)
%t A265613 Table[(4^(n + 1) n (n + 1) (3 n + 2) Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 5]), {n, 0, 25}] (* or *)
%t A265613 Table[CatalanNumber(n+1) n (3 n^2 + 5 n + 2)/((4 + n) (3 + n)), {n, 0, 25}] (* _Michael De Vlieger_, Dec 15 2015 *)
%o A265613 (Sage)
%o A265613 a = lambda n: catalan_number(n+1)*n*(3*n^2+5*n+2)/((4+n)*(3+n))
%o A265613 [a(n) for n in range(26)]
%Y A265613 Cf. A000108, A007946, A097613, A051960, A029651, A051924, A129869, A220101, A264357, A265612.
%K A265613 nonn
%O A265613 0,3
%A A265613 _Peter Luschny_, Dec 15 2015