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A277956 a(n) = (n+2)*Sum_{i=0..n}(binomial(3*n-2*i+1, n-i)/(2*n-i+2)).

%I A277956 #20 Apr 10 2017 23:00:31
%S A277956 1,4,19,101,573,3382,20483,126292,788878,4976489,31635811,202354517,
%T A277956 1300880374,8398175713,54409200963,353571026085,2303666554659,
%U A277956 15043760670031,98439176169692,645290365460761,4236768489465944,27857102370774193
%N A277956 a(n) = (n+2)*Sum_{i=0..n}(binomial(3*n-2*i+1, n-i)/(2*n-i+2)).
%H A277956 G. C. Greubel, <a href="/A277956/b277956.txt">Table of n, a(n) for n = 0..1000</a>
%F A277956 G.f.: F'(x)*F(x)/(1-F(x))/x, where F(x)/x is g.f. of A001764.
%F A277956 From _Vaclav Kotesovec_, Nov 06 2016: (Start)
%F A277956 Recurrence: 2*(n+1)*(2*n + 1)*(91*n^4 - 232*n^3 + 15*n^2 + 266*n - 120)*a(n) = (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-1) - (2821*n^6 - 4189*n^5 - 10027*n^4 + 18573*n^3 - 3498*n^2 - 3968*n + 960)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(91*n^4 + 132*n^3 - 135*n^2 - 36*n + 20)*a(n-3).
%F A277956 a(n) ~ 3^(3*n+7/2) / (7 * sqrt(Pi*n) * 2^(2*n+3)). (End)
%F A277956 a(n) = A026004(n)*hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4). - _Peter Luschny_, Nov 06 2016
%p A277956 h := n -> hypergeom([1,-2*n-2,-n],[-3*n/2-1/2,-3*n/2],1/4):
%p A277956 b := n -> binomial(3*n+1,n)*(n+2)/(2*n+2): # A026004
%p A277956 a := n -> `if`(n=0,1,b(n)*simplify(h(n))):
%p A277956 seq(a(n), n=0..21); # _Peter Luschny_, Nov 06 2016
%t A277956 f[n_] := (n + 2)Sum[ Binomial[3n - 2i + 1, n - i]/(2n - i + 2), {i, 0, n}]; Array[f, 22, 0] (* _Robert G. Wilson v_, Nov 06 2016 *)
%o A277956 (Maxima)
%o A277956 F(x):=x*(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)));
%o A277956 taylor(diff(F(x),x)*F(x)/(1-F(x))/x,x,0,10);
%o A277956 (PARI) for(n=0,25, print1((n+2)*sum(i=0,n,(binomial(3*n-2*i+1, n-i)/(2*n-i+2))), ", ")) \\ _G. C. Greubel_, Apr 09 2017
%Y A277956 Cf. A001764, A026004, A242798.
%K A277956 nonn
%O A277956 0,2
%A A277956 _Vladimir Kruchinin_, Nov 05 2016